3.1 Construction of the approximate solution
We denote by ε the smallest positive parameter satisfying
$$\begin{aligned} \rho ^{4} = \frac{384 \varepsilon ^{4}}{(1+\varepsilon ^{2})^{4}}. \end{aligned}$$
Let
$$\begin{aligned} u_{\varepsilon} (x): = 4 \ln \bigl(1+{\varepsilon }^{2} \bigr)- 4 \ln \bigl(\varepsilon ^{2}+ \vert x \vert ^{2} \bigr), \end{aligned}$$
(33)
which is a solution of
$$\begin{aligned} \Delta ^{2} u =\rho ^{4}e^{u} \quad\text{in } \mathbb{R}^{4}. \end{aligned}$$
(34)
Hence for all \(\tau > 0 \) the function
$$\begin{aligned} u_{\varepsilon,\tau} (x): = 4 \ln \bigl(1+{\varepsilon }^{2} \bigr) + 4 \ln \tau -4\ln \bigl(\varepsilon ^{2}+ \vert \tau x \vert ^{2} \bigr) \end{aligned}$$
(35)
is also solution to (34).
3.1.1 A linearized operator
First we introduce some definitions and notations:
Definition 1
Given \(k\in {\mathbb{N}}\), \(\alpha \in (0,1)\), \(\mu \in {\mathbb{R}}\), and \(|x|=r\), we define the Hölder weighted space \({\mathcal{C}}^{k,\alpha}_{\mu}(\mathbb{R}^{4})\) as the space of functions \(w \in {\mathcal{C}}^{k,\alpha}_{\mathrm{loc}}(\mathbb{R}^{4})\) for which the following norm
$$\begin{aligned} \Vert u \Vert _{{\mathcal{C}}^{k,\alpha}_{\mu}(\mathbb{R}^{4})} = \Vert u \Vert _{{\mathcal{C}}^{k, \alpha}(\bar{B}_{1}(0))} + {\sup _{r\geq 1}} \bigl( \bigl(1+r^{2} \bigr)^{-{ \frac{\mu}{2}}} \bigl\Vert u(r \cdot ) \bigr\Vert _{{\mathcal{C}}^{k,\alpha} (\bar{B}_{1}(0)-B_{{\frac{1}{2}}}(0) )} \bigr) \end{aligned}$$
is finite. Similarly, for given \(\bar{r} \geq 1\), let \({\mathcal{C}}^{k,\alpha}_{\mu}(B_{\bar{r}}(0))\) be the space of functions in \({\mathcal{C}}^{k,\alpha}(B_{\bar{r}}(0))\) for which the following norm
$$\begin{aligned} \Vert u \Vert _{{\mathcal{C}}^{k,\alpha}_{\mu}(B_{\bar{r}}(0))} = \Vert u \Vert _{{\mathcal{C}}^{k, \alpha}( B_{1}(0))} + \sup _{ 1 \leq r\leq \bar{r}} \bigl( r^{-\mu} \bigl\Vert u(r \cdot ) \bigr\Vert _{{\mathcal{C}}^{k,\alpha} ( \bar{B}_{1}(0) - B_{{\frac{1}{2}}}(0) )} \bigr) \end{aligned}$$
is finite. Finally, set \(B^{*}_{r}(x^{i}) = B_{r}(x^{i}) - \{x^{i}\}\), let \({\mathcal{C}}^{k,\alpha}_{\mu}(\bar{B}_{1}^{*}(0))\) be the space of functions in \({\mathcal{C}}^{k,\alpha}_{\mathrm{loc}}(\bar{B}_{1}^{*}(0))\) for which the following norm
$$\begin{aligned} \Vert u \Vert _{{\mathcal{C}}^{k,\alpha}_{\mu}(\bar{B}_{1}^{*}(0))} = { \sup_{r\leq {\frac{1}{2}}}} \bigl( r^{-\mu} \bigl\Vert u(r \cdot ) \bigr\Vert _{{ \mathcal{C}}^{k,\alpha}(\bar{B}_{2}(0)-B_{1}(0))} \bigr) \end{aligned}$$
is finite.
We define the linear elliptic operator \(\mathbbm{{}\mathbb{L}\mathbbm{}}\) by
$$\begin{aligned} \mathbbm{{}\mathbb{L}\mathbbm{}}: = \Delta ^{2} - \frac{384}{(1+r^{2})^{4}}, \end{aligned}$$
which is the linearized operator of \(\Delta ^{2} u - \rho ^{4}e^{u}=0\) about the radial symmetric solution \(u_{\varepsilon =1, \tau =1}\) defined by (35). When \(k \geq 2\), we let \([ {\mathcal{C}}^{k,\alpha}_{\mu }(\bar{\Omega}) ]_{0}\) to be the subspace of functions \(w \in {\mathcal{C}}^{k,\alpha}_{\mu }(\bar{\Omega})\) satisfying \(\Delta w = w = 0\) on ∂Ω.
For all \(\varepsilon, \lambda, \tau _{i} >0\), \(i=1,2,3 \) and \(\gamma, \xi \in (0, 1)\), we define
$$\begin{aligned} r_{\varepsilon, \lambda}:= \max \bigl(\varepsilon ^{{\frac{1}{2}}}, \lambda ^{{\frac{1}{2}}}, \varepsilon ^{\frac{\gamma +\xi -1}{\gamma}}, \varepsilon ^{\frac{\gamma +\xi -1}{\xi}} \bigr)\quad \text{and}\quad R^{i}_{\varepsilon, \lambda}:= \tau _{i} \frac {r_{\varepsilon, \lambda}}{\varepsilon }. \end{aligned}$$
(36)
Proposition 1
[8] All bounded solutions of \(\mathbbm{{}\mathbb{L}\mathbbm{}}w =0\) on \(\mathbbm{{}\mathbb{R}\mathbbm{}}^{4} \) are linear combination of
$$\begin{aligned} \phi _{0}(x) = 4 \frac{1 - \vert x \vert ^{2}}{1 + \vert x \vert ^{2}} \quad\textit{and}\quad \phi _{i}(x) = \frac{8 x_{i}}{1 + \vert x \vert ^{2}}\quad \textit{for } i = 1, \ldots, 4. \end{aligned}$$
Moreover, for \(\mu > 1\), \(\mu \notin \mathbb{Z}\), the operator \(\mathbb{L}: {\mathcal {C}}^{4,\alpha}_{\mu}(\mathbb{R}^{4}) \longrightarrow {\mathcal {C}}^{0,\alpha}_{\mu -4}(\mathbb{R}^{4})\) is surjective.
In the following, we denote by \({\mathcal{G}}_{\mu}\) to be a right inverse of \(\mathbb{L}\). Similarly, using the fact that any bounded bi-harmonic solution on \(\mathbbm{{}\mathbb{R}\mathbbm{}}^{4} \) is constant, we claim
Proposition 2
Let \(\delta > 0\), \(\delta \notin \mathbb{Z}\) then \(\Delta ^{2}\) is surjective from \({\mathcal {C}}^{4,\alpha}_{\delta}(\mathbb{R}^{4})\) to \({\mathcal {C}}^{0,\alpha}_{\delta -4}(\mathbb{R}^{4})\).
We denote by \({{\mathcal {K}}}_{\delta}: {\mathcal {C}}^{0,\alpha}_{ \delta -4}(\mathbb{R}^{4}) \longrightarrow {\mathcal {C}}^{4, \alpha}_{\delta}(\mathbb{R}^{4}) \) a right inverse of \(\Delta ^{2}\) for \(\delta > 0\), \(\delta \notin \mathbb{Z}\).
Finally, we consider punctured domains. Given \(\tilde{x}^{1}, \tilde{x}^{2}, \tilde{x}^{3}\) three distinct points in Ω, we define \({\tilde{\mathbf {x}}}:=(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3}) \) and \(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}):= \bar{\Omega}- \{ \tilde{x}^{1}, \tilde{x}^{2},\tilde{x}^{3} \}\). Let \(r_{0} > 0 \) be small such that \(\bar{B}_{r_{0}}(\tilde{x}^{i})\) are disjoint and included in Ω. For all \(r \in (0,r_{0}) \), we define
$$\begin{aligned} \bar{\Omega}_{r}({\tilde{\mathbf {x}}}):= \bar{\Omega}- { \bigcup _{i=1}^{3}} B_{r} \bigl( \tilde{x}^{i} \bigr). \end{aligned}$$
Definition 2
Let \(k\in \mathbb{R}, \alpha \in (0,1) \) and \(\nu \in \mathbb{R} \), we introduce the Hölder weighted space \({\mathcal {C}}^{k, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) \) as the space of functions \(w \in {\mathcal {C}}_{\mathrm{loc}}^{k, \alpha}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))\) such that
$$\begin{aligned} \Vert w \Vert _{{\mathcal {C}}^{k, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )}:= \Vert w \Vert _{{\mathcal {C}}^{k, \alpha}(\bar{\Omega}_{\frac{r_{0}}{2}}({\tilde{\mathbf {x}}}))} + \sum _{i=1}^{3} {\sup_{0 < r \leq \frac{r_{0}}{2} }} \bigl( r^{-\nu} \bigl\Vert w \bigl( \tilde{x}^{i} + r. \bigr) \bigr\Vert _{{\mathcal {C}}^{k, \alpha} (\bar{B}_{2}(0)-B_{1}(0))} \bigr) \end{aligned}$$
is finite.
Furthermore, for \(k \geq 2 \), let \([{\mathcal {C}}^{k, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) ]_{0}\) to be the set of \(w \in {\mathcal {C}}^{k, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))\) satisfying \(\Delta w = w = 0 \) on ∂Ω.
We recall the following result.
Proposition 3
[17] Let \(\nu <0 \), \(\nu \notin \mathbb{Z}\), then \(\Delta ^{2}\) is surjective from \([{\mathcal {C}}^{4,\alpha}_{\nu}(\bar{\Omega}^{*}({ \tilde{\mathbf {x}}})) ]_{0}\) to \({\mathcal {C}}^{0,\alpha}_{\nu -4}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))\).
We denote by \(\tilde{{{\mathcal {G}}}}_{\nu}: {\mathcal {C}}^{0,\alpha}_{ \nu -4}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) \longrightarrow [ {\mathcal {C}}^{4,\alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) ]_{0}\) a right inverse of \(\Delta ^{2}\) for \(\nu <0 \), \(\nu \notin \mathbb{Z}\).
3.1.2 Ansatz and first estimates
For all \(\sigma \geq 1 \), we denote by \({\xi}_{\mu,\sigma}: {\mathcal {C}}^{0, \alpha}_{\mu}(\bar{B}_{\sigma}(0)) \longrightarrow {\mathcal {C}}^{0,\alpha}_{\mu }({\mathbb{R}}^{4})\) the extension operator defined by
$$\begin{aligned} \textstyle\begin{cases} {\xi}_{\mu,\sigma} (f)(x)\equiv f(x) &\text{for } \vert x \vert \leq \sigma, \\ {\xi}_{\mu,\sigma}(f)(x) = \chi ( \frac{ \vert x \vert }{\sigma} ) f ( \sigma \frac{x}{ \vert x \vert } ) & \text{for } \vert x \vert \geq \sigma. \end{cases}\displaystyle \end{aligned}$$
(37)
Here χ is a cut-off function over \(\mathbb{R}_{+}\), which is equal to 1 for \(t\leq 1 \) and equal to 0 for \(t\geq 2 \). It is easy to check that there exists a constant \(c =\bar{c}(\mu ) > 0 \), independent of σ such that
$$\begin{aligned} \bigl\Vert \xi _{\mu,\sigma}(w) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\mu}(\mathbb{R}^{4})} \leq \bar{c} \Vert w \Vert _{{\mathcal {C}}^{0, \alpha}_{\mu}(\bar{B}_{ \sigma}(0))}. \end{aligned}$$
(38)
Here, we are interested to study the system
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} u_{1} + \mathcal{L}_{\lambda}(u_{1}) = \rho ^{4} e^{ \gamma u_{1} + (1-\gamma ) u_{2}}, \\ \Delta ^{2} u_{2} + \mathcal{L}_{\lambda}(u_{2}) = \rho ^{4} e^{ \xi u_{2} + (1-\xi ) u_{1}}, \end{cases}\displaystyle \end{aligned}$$
(39)
where
$$\begin{aligned} \mathcal{L}_{\lambda}(u_{i})=\lambda (\Delta u_{i})^{2} + \lambda \nabla u_{i} \cdot \nabla ( \Delta u_{i})+ \lambda ^{2} \vert \nabla u_{i} \vert ^{2} \Delta u_{i}, \quad\text{for } i=1,2. \end{aligned}$$
(40)
Using the following transformations
$$\begin{aligned} &\textstyle\begin{cases} v_{1}(x) = u_{1}(\frac{\varepsilon }{\tau _{1}}x)+ \frac{8}{\gamma} \ln{\varepsilon} - \frac{4}{\gamma} \ln ({\frac{\tau _{1}(1+\varepsilon ^{2})}{2}}) &\text{in } B_{r_{\varepsilon, \lambda}}(x^{1}), \\ v_{2}(x) = u_{2}(\frac{\varepsilon }{\tau _{1}}x)& \text{in } B_{r_{\varepsilon, \lambda}}(x^{1}), \end{cases}\displaystyle \end{aligned}$$
(41)
$$\begin{aligned} &\textstyle\begin{cases} v_{1}(x) = u_{1}(\frac{\varepsilon }{\tau _{2}}x)+ 8 \ln{ \varepsilon} - 4 \ln ({\frac{\tau _{2}(1+\varepsilon ^{2})}{2}}) & \text{in } B_{r_{\varepsilon, \lambda}}(x^{2}), \\ v_{2}(x) = u_{2}(\frac{\varepsilon }{\tau _{2}}x)+ 8 \ln{ \varepsilon} - 4 \ln ({\frac{\tau _{2}(1+\varepsilon ^{2})}{2}}) & \text{in } B_{r_{\varepsilon, \lambda}}(x^{2}) \end{cases}\displaystyle \end{aligned}$$
(42)
and
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x) = u_{1}(\frac{\varepsilon }{\tau _{3}}x)& \text{in } B_{r_{\varepsilon, \lambda}}(x^{3}), \\ v_{2}(x) = u_{2}(\frac{\varepsilon }{\tau _{3}}x)+ \frac{8}{\xi} \ln{\varepsilon} - \frac{4}{\xi} \ln ({\frac{\tau _{3}(1+\varepsilon ^{2})}{2}}) & \text{in } B_{r_{\varepsilon, \lambda}}(x^{3}). \end{cases}\displaystyle \end{aligned}$$
(43)
Thus, the previous systems can be written as
$$\begin{aligned} &\textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1})= 24 e^{\gamma v_{1} + (1-\gamma ) v_{2}} &\text{in } B_{R_{\varepsilon, \lambda}^{1}}(x^{1}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2}) = 24 C_{1,\varepsilon }^{4\frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} e^{\xi v_{2} +(1-\xi ) v_{1}} &\text{in } B_{R_{\varepsilon, \lambda}^{1}}(x^{1}), \end{cases}\displaystyle \end{aligned}$$
(44)
$$\begin{aligned} &\textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1}) = 24 e^{\gamma v_{1} + (1-\gamma ) v_{2}} &\text{in } B_{R_{\varepsilon, \lambda}^{2}}(x^{2}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2})= 24 e^{\xi v_{2} +(1- \xi ) v_{1}} &\text{in } B_{R_{\varepsilon, \lambda}^{2}}(x^{2}), \end{cases}\displaystyle \end{aligned}$$
(45)
and
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1})= 24 C_{3, \varepsilon }^{4\frac{\gamma +\xi -1}{\xi}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\xi}} e^{\gamma v_{1} + (1-\gamma ) v_{2}} & \text{in } B_{R_{\varepsilon, \lambda}^{3}}(x^{3}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2})= 24 e^{\xi v_{2} +(1-\xi ) v_{1}} &\text{in } B_{R_{\varepsilon, \lambda}^{3}}(x^{3}), \end{cases}\displaystyle \end{aligned}$$
(46)
where \(C_{i,\varepsilon }={\frac{2}{\tau _{i} (1+\varepsilon ^{2})}}\) for \(i=1,3\). Here \(\tau _{i} > 0\) is a constant which will be fixed later.
We denote by \(\bar{u}=u_{\varepsilon =1,\tau _{i}=1}\), we look for a solution of (44) of the form
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x) = {\frac{1}{\gamma}}\bar{u}(x-x^{1}) - {\frac{1-\gamma}{\gamma}} G(\frac{\varepsilon x}{\tau _{1}}, x^{2})- {\frac{1-\gamma}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})- \frac{\ln \gamma}{\gamma} + h^{1}_{1}(x), \\ v_{2}(x) = {\frac{1}{\xi}}G(\frac{\varepsilon x}{\tau _{1}}, x^{3}) +G( \frac{\varepsilon x}{\tau _{1}}, x^{2}) + h^{1}_{2}(x). \end{cases}\displaystyle \end{aligned}$$
(47)
Using the fact that \(e^{\bar{u}(x-x^{1})}=\frac{16}{(1+|x-x^{1}|^{2})^{4}}\), we see that this amounts to solve the system
$$\begin{aligned} \textstyle\begin{cases} {\mathbb{L}} h^{1}_{1} = \frac{384 }{\gamma (1 + r^{2})^{4}} (e^{ \gamma h^{1}_{1}+(1-\gamma )h^{1}_{2}} - \gamma h^{1}_{1} -1 )- \mathcal{L}_{\lambda} ({\frac{1}{\gamma}}\bar{u}(x-x^{1}) - {\frac{1- \gamma}{\gamma}} G(\frac{\varepsilon x}{\tau _{1}},x^{2}) \\ \phantom{{\mathbb{L}} h^{1}_{1} =}{} - {\frac{1-\gamma}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{1}}, x^{3}) - \frac{\ln \gamma}{\gamma} + h^{1}_{1}(x) ), \\ \Delta ^{2} h^{1}_{2} = { \frac{24 C_{1,\varepsilon }^{4\frac{\gamma +\xi -1}{\gamma }} 16^{\frac{1-\xi}{\gamma}} \varepsilon ^{8\frac{\gamma +\xi -1}{\gamma}}}{\gamma ^{\frac{1-\xi}{\gamma}}(1+r^{2})^{4{\frac{1-\xi}{\gamma}}}} e^{{\frac{\gamma +\xi -1}{\gamma}}G(\frac{\varepsilon x}{\tau _{1}}, x^{2})+{ \frac{\gamma + \xi -1}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})+ \xi h^{1}_{2} +(1-\xi ) h^{1}_{1}}} \\ \phantom{\Delta ^{2} h^{1}_{2} =}{}- \mathcal{L}_{\lambda} ({\frac{1}{\xi}}G( \frac{\varepsilon x}{\tau _{1}}, x^{3})+G( \frac{\varepsilon x}{\tau _{1}}, x^{2}) + h^{1}_{2}(x) ), \end{cases}\displaystyle \end{aligned}$$
(48)
We denote by
$$\begin{aligned} {\mathbb{L}} h^{1}_{1} = \mathcal{R}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr)\quad \text{and}\quad \Delta ^{2} h^{1}_{2} = \mathcal{R}_{2} \bigl(h_{1}^{1},h_{2}^{1} \bigr). \end{aligned}$$
Fix \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma + \xi -1}{\xi}})\} )\). To find a solution of (48), it is enough to find a fixed point \((h_{1}^{1},h_{2}^{1})\) in a small ball of \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4}) \times \mathcal{C}^{4, \alpha}_{\delta}(\mathbb{R}^{4})\) solutions of
$$\begin{aligned} \textstyle\begin{cases} h_{1}^{1} = \mathcal {G}_{\mu }\circ{\xi}_{\mu, R_{\varepsilon, \lambda}^{1}} \circ \mathcal{R}_{1}(h_{1}^{1},h_{2}^{1}) = \mathcal{N}_{1}(h_{1}^{1},h_{2}^{1}), \\ h_{2}^{1} = {\mathcal {K}}_{\delta }\circ{\xi}_{\delta, R_{ \varepsilon, \lambda}^{1}} \circ \mathcal{R}_{2}(h_{1}^{1},h_{2}^{1})= \mathcal{M}_{1}(h_{1}^{1},h_{2}^{1}). \end{cases}\displaystyle \end{aligned}$$
(49)
Here \(\xi _{\mu,R_{\varepsilon, \lambda}^{1}}\) is defined in (37), \({\mathcal {G}}_{\mu}\) and \({\mathcal {K}}_{\delta}\) are defined after Propositions 1, 2, respectively. Then we have the following result.
Lemma 1
Given \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0 \), \(\bar{c}_{\kappa} > 0 \) and \(\gamma _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0},1)\), \(\mu \in (1,2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma + \xi -1}{\xi}})\} )\). We have
$$\begin{aligned} &\bigl\Vert \mathcal{ N}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}, \qquad\bigl\Vert \mathcal{ M}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4} )} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2},\\ &\bigl\Vert \mathcal{ N}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr) -\mathcal{ N }_{1} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\leq \bar{c}_{ \kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} +\bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{1}-k_{2}^{1} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4} )} \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert \mathcal{ M}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr) -\mathcal{ M }_{1} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq \bar{c}_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert \bigl(h_{1}^{1},h_{2}^{1} \bigr)- \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}, \end{aligned}$$
provided \((h_{1}^{1}, h_{2}^{1}), (k_{1}^{1}, k_{2}^{1}) \in {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{ \delta }(\mathbb{R}^{4})\) satisfying
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2},\qquad \bigl\Vert \bigl(k_{1}^{1} , k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4}) \times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{ \kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(50)
Proof
Using the fact that \(\vert \nabla ^{i} G(x,y) \vert \leq c\vert x-y\vert ^{-i}\) for \(i \geq 1\), we get
$$\begin{aligned} &\sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \mathcal{R}_{1}(0,0) \bigr\vert \\ &\quad\leq \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \mathcal{L}_{\lambda} \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1-\gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} \biggr) \biggr\vert \\ &\quad\leq \lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \biggl\vert \biggl(\Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - { \frac{1- \gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + 2\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - { \frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \nabla \biggl(\Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1- \gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \biggr) \biggr) \biggr\vert \\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{}\times \biggl\vert \Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \biggr) \biggr\vert \\ & \quad\leq c_{\kappa}\lambda \bigl(1+ \varepsilon + \varepsilon ^{4} R_{ \varepsilon, \lambda}^{4-\mu}+ \varepsilon ^{2} R_{\varepsilon, \lambda}^{2-\mu}+ \varepsilon ^{3} R_{\varepsilon, \lambda}^{3-\mu} \bigr)+ c_{\kappa}\lambda ^{2} \bigl(1+ \varepsilon ^{2} R_{\varepsilon, \lambda}^{3- \mu}+ \varepsilon R_{\varepsilon, \lambda}^{2-\mu}+ \varepsilon ^{3} R_{ \varepsilon, \lambda}^{4-\mu} \bigr) \\ &\quad\leq c_{\kappa}\lambda \bigl(1+ \varepsilon + \varepsilon ^{\mu} r_{ \varepsilon, \lambda}^{4-\mu} \bigr)+ c_{\kappa}\lambda ^{2} \bigl(1+ \varepsilon ^{\mu -1} r_{\varepsilon, \lambda}^{4-\mu} \bigr). \end{aligned}$$
Making use of Proposition 1 together with (38), for \(\mu \in (1,2)\), we get that there exists \(c_{\kappa} > 0\) such that
$$\begin{aligned} \bigl\Vert \mathcal{N}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(51)
For the second estimate, we have
$$\begin{aligned} &\sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \bigl\vert \mathcal{R}_{2}(0,0) \bigr\vert \\ &\quad \leq c_{\kappa }\sup_{r \leq R_{\varepsilon, \lambda}^{1}} C_{1, \varepsilon }^{4\frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} r^{4-\delta} \biggl( \frac{16}{(1+r^{2})^{4}} \biggr)^{ \frac{1 - \xi}{\gamma} } e^{ \frac{\gamma + \xi - 1}{\gamma} G(\frac{\varepsilon x}{\tau _{1}}, x^{2}) +\frac{\gamma + \xi - 1}{\gamma \xi } G( \frac{\varepsilon x}{\tau _{1}}, x^{3})} \\ &\qquad{} + \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \mathcal{L}_{\lambda} \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \\ & \quad\leq c_{\kappa }C_{1,\varepsilon }^{4\frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8\frac{\gamma + \xi -1}{\gamma }} \sup_{r \leq R_{ \varepsilon, \lambda}^{1}} S(r) +\lambda \sup _{r \leq R_{ \varepsilon, \lambda}^{1}} r^{4-\delta} \biggl\vert \Delta \biggl({ \frac{1}{\xi}}G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{} + \lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta} \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \biggr\vert \\ &\qquad{}\times\biggl\vert \nabla \biggl(\Delta \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \biggr) \biggr\vert \\ &\qquad{}+ \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta} \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{}\times\biggl\vert \Delta \biggl({ \frac{1}{\xi}}G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr) \biggr\vert , \end{aligned}$$
where \(S(r) = \frac{r^{4-\delta}}{(1 + r^{2})^{4{\frac{1-\xi}{\gamma}}}}\). Then, using the fact that \(\vert \nabla ^{i} G(x,y) \vert \leq c\vert x-y\vert ^{-i}\) for \(i \geq 1\), we get
$$\begin{aligned} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \bigl\vert \mathcal{R}_{2}(0,0) \bigr\vert & \leq c_{\kappa }C_{1,\varepsilon }^{4 \frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8 \frac{\gamma + \xi -1}{\gamma }} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} S(r) +c_{\kappa } \lambda \varepsilon ^{4} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} +c_{\kappa }\lambda ^{2} \varepsilon ^{4} \sup _{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta}. \end{aligned}$$
If \(4 - \delta - 8 {\frac{1-\xi}{\gamma}} \leq 0 \), then S is bounded on \(\mathbb{R}_{+}\). If \(4-\delta -8\frac{1-\xi}{\gamma} > 0, \sup_{[0, \frac{r_{\varepsilon, \lambda}}{\varepsilon }[} S(r) = S ( \frac{r_{\varepsilon, \lambda}}{\varepsilon } ) \), then we get
$$\begin{aligned} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \bigl\vert \mathcal{R}_{2}(0,0) \bigr\vert &\leq c_{\kappa }\max \bigl\{ \varepsilon ^{8 \frac{\gamma + \xi -1}{\gamma }}, \varepsilon ^{\delta +4} r_{ \varepsilon, \lambda}^{4- \delta -8\frac{1-\xi}{\gamma}} \bigr\} + c_{ \kappa }\lambda \varepsilon ^{\delta} r_{\varepsilon, \lambda}^{4- \delta}+ c_{\kappa }\lambda ^{2} \varepsilon ^{\delta} r_{ \varepsilon, \lambda}^{4- \delta} \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
Using the same argument as above, we get \(\|\mathcal{M}_{1}(0,0) \|_{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})}\leq c_{\kappa }r_{\varepsilon, \lambda}^{2}\).
Recall the following conditions.
$$\begin{aligned} &(A_{1})\quad \text{If } 0 < \varepsilon < \lambda, \text{ then } \lambda ^{1+\frac{\mu}{2}} \varepsilon ^{-\mu}\rightarrow 0 \text{ as } \lambda \rightarrow 0, \text{ for any } \mu \in (1,2). \\ &(A_{2}) \quad\text{If } 0 < \varepsilon < \lambda, \text{ then } \lambda ^{1+\frac{\delta}{2}} \varepsilon ^{-\delta}\rightarrow 0 \text{ as } \lambda \rightarrow 0, \text{ for any } \delta \in \biggl(0, \min \biggl\{ \biggl({ \frac{\gamma +\xi -1}{\gamma}} \biggr), \\ &\phantom{(A_{2}) \quad}\biggl({\frac{\gamma +\xi -1}{\xi}} \biggr) \biggr\} \biggr). \end{aligned}$$
To derive the third estimate, using the fact that for all functions in \({\mathcal {C}}_{\mu}^{k,\alpha}(\mathbb{R}^{4})\) bounded by a constant times \((1+r^{2})^{\mu /2}\) have their l-th partial derivatives that are bounded by \((1+r^{2})^{(\mu -l)/2}\), for \(l=1,\ldots,k+\alpha,\ldots \) (a.e. \(\vert \nabla ^{l}w \vert \leq c_{\kappa }r^{\mu - l} \Vert w\Vert _{{ \mathcal {C}}_{\mu}^{k,\alpha}(\mathbb{R}^{4})}\), \((1+r^{2})^{(\mu -l)/2} \sim r^{\mu -l}\) for r very large) and the fact that \(\vert \nabla ^{i} G(x,y) \vert \leq c\vert x-y\vert ^{-i}\) for \(i \geq 1\). Then for \((h_{1}^{1},h_{2}^{1}), (k_{1}^{1},k_{2}^{1})\) verifying (50), we have
$$\begin{aligned} &\sup_{r\leq R_{\varepsilon, \lambda}^{1}} r^{4 - \mu} \bigl\vert \mathcal{R}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr)-\mathcal{R}_{1} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\vert \\ &\quad\leq \sup_{r\leq R_{\varepsilon, \lambda}^{1}} \frac{384r^{4-\mu}}{(1+r^{2})^{4}}\frac{1}{\gamma} \bigl\vert \bigl( e^{ \gamma h_{1}^{1} + (1 - \gamma ) h_{2}^{1} } - \gamma h_{1}^{1} - 1 \bigr) - \bigl( e^{\gamma k_{1}^{1}+ (1 - \gamma ) k_{2}^{1}} - \gamma k_{1}^{1} - 1 \bigr) \bigr\vert \\ &\qquad{}+ \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \mathcal{L}_{\lambda} \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr)\\ &\qquad{} - {\frac{1-\gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} +h^{1}_{1}(x) \biggr) \\ &\qquad{} - \mathcal{L}_{\lambda} \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1- \gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} +k^{1}_{1}(x) \biggr) \biggr\vert \\ &\quad\leq c_{\kappa} \sup_{r\leq R_{\varepsilon, \lambda}^{1}} \frac{384 r^{4-\mu}}{(1+r^{2})^{4}} \frac{1}{\gamma} \bigl[ \gamma ^{2} \bigl( \bigl(h_{1}^{1} \bigr)^{2}- \bigl(k_{1}^{1} \bigr)^{2} \bigr) + (1 - \gamma ) \bigl\vert h_{2}^{1} - k_{2}^{1} \bigr\vert \bigr]\\ &\qquad{} +\lambda \sup _{r\leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \bigl\vert \Delta \bigl(h^{1}_{1}-k^{1}_{1} \bigr) \bigr\vert \\ &\qquad{}\times \biggl(\frac{2}{\gamma} \vert \Delta {\bar{u}} \vert + 2{ \frac{1-\gamma}{\gamma}} \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert + 2{\frac{1- \gamma}{\gamma \xi}} \biggl\vert \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert + \bigl\vert \Delta h^{1}_{1} \bigr\vert + \bigl\vert \Delta k^{1}_{1} \bigr\vert \biggr) \\ &\qquad{}+\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \nabla \bigl(h^{1}_{1}-k^{1}_{1} \bigr) \cdot \nabla \biggl( \frac{2}{\gamma} \Delta {\bar{u}} + 2{\frac{1-\gamma}{\gamma}} \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}+ 2{ \frac{1-\gamma}{\gamma \xi}} \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + \Delta h^{1}_{1} + \Delta k^{1}_{1} \biggr) \\ &\qquad{} +\nabla \bigl(\Delta \bigl(h^{1}_{1}-k^{1}_{1} \bigr) \bigr) \cdot \nabla \biggl( \frac{2}{\gamma} {\bar{u}} - 2{ \frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- 2{ \frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + h^{1}_{1}(x) + k^{1}_{1}(x) \biggr) \biggr\vert \\ &\qquad{} +\lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \bigl\vert \Delta \bigl(h^{1}_{1}-k^{1}_{1} \bigr) \bigr\vert \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\gamma}} \bar{u} - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +h^{1}_{1}(x) \biggr) \biggr\vert ^{2} \\ & \qquad{}+ \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} - { \frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +k^{1}_{1}(x) \biggr) \biggr\vert ^{2} \biggr] \\ &\qquad{}+ \lambda ^{2} \sup_{r \leq R_{ \varepsilon, \lambda}^{1}} r^{4-\mu} \biggl( \frac{2}{\gamma} \vert \Delta {\bar{u}} \vert + 2{\frac{1-\gamma}{\gamma}} \biggl\vert \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert + 2{\frac{1-\gamma}{\gamma \xi}} \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert + \bigl\vert \Delta h^{1}_{1} \bigr\vert \\ &\qquad{}+ \bigl\vert \Delta k^{1}_{1} \bigr\vert \biggr) \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{} - {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +h^{1}_{1}(x) \biggr) \biggr\vert ^{2} \\ &\qquad{}- \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1-\gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +k^{1}_{1}(x) \biggr) \biggr\vert ^{2} \biggr] \\ &\quad\leq c_{\kappa} \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \bigr) \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{1}-k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} \\ &\qquad{}+ {c}_{\kappa } \lambda \bigl( 1 + \varepsilon R_{\varepsilon, \lambda}+ \varepsilon ^{2} R_{ \varepsilon, \lambda}^{2} + \varepsilon ^{3} R_{\varepsilon, \lambda}^{3} \\ &\qquad{} + R_{\varepsilon, \lambda}^{\mu } \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \bigr) \bigr) \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}}\\ &\qquad{} + {c}_{\kappa} \lambda ^{2} \bigl( 1 + \varepsilon R_{\varepsilon, \lambda}+ \varepsilon ^{2} R_{ \varepsilon, \lambda}^{2} + \varepsilon ^{3} R_{\varepsilon, \lambda}^{3} + R_{\varepsilon, \lambda}^{\mu } \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \bigr) \\ &\qquad{} + \varepsilon R_{\varepsilon, \lambda}^{\mu +1} \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} \bigr)+ \varepsilon ^{2} R_{\varepsilon, \lambda}^{\mu +2} \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \bigr) \\ &\qquad{}+ \varepsilon ^{2} R_{ \varepsilon, \lambda}^{2\mu} \bigl( \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} + \bigl\Vert k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \bigr)^{2} \bigr) \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \\ &\quad\leq c_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{1}-k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}}\\ &\qquad{} + {c}_{\kappa } \lambda \bigl( 1 + r_{\varepsilon, \lambda} + \varepsilon ^{-\mu} r_{ \varepsilon, \lambda}^{2+\mu} \bigr) \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} \\ &\qquad{} + {c}_{\kappa} \lambda ^{2} \bigl( 1 + r_{ \varepsilon, \lambda} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+ \mu}+ \varepsilon ^{-2\mu} r_{\varepsilon, \lambda}^{2(2+\mu )} \bigr) \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}. \end{aligned}$$
Using the following estimates
$$\begin{aligned} c_{\kappa }\varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} \leq \textstyle\begin{cases} c_{\kappa }\varepsilon ^{1 - \frac{\mu}{2}} &\text{for } \varepsilon > \lambda, \\ c_{\kappa }\lambda ^{1 + \frac{\mu}{2}}\varepsilon ^{-\mu} & \text{for } \lambda > \varepsilon, \end{cases}\displaystyle \end{aligned}$$
together with condition \((A_{1})\), yield \({c}_{\kappa } \lambda ( 1+ r_{\varepsilon, \lambda} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} ) \leq {c}_{ \kappa }r_{\varepsilon, \lambda}^{2}\) and \({c}_{\kappa }\lambda ^{2} ( 1+ r_{\varepsilon, \lambda} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} + \varepsilon ^{-2 \mu} r_{\varepsilon, \lambda}^{2(2+\mu )} )\leq {c}_{\kappa }r_{ \varepsilon, \lambda}^{2}\). Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get that there exists \(\bar{c}_{\kappa} > 0\) such that
$$\begin{aligned} &\bigl\Vert \mathcal{ N}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr) -\mathcal{ N }_{1} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad\leq \bar{c}_{ \kappa } r_{\varepsilon, \lambda}^{2} \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{1}-k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(52)
On the other hand, we have
$$\begin{aligned} & \sup_{r\leq R_{\varepsilon, \lambda}^{1}} r^{4 - \delta} \bigl\vert \mathcal{R}_{2} \bigl(h_{1}^{1},h_{2}^{1} \bigr) -\mathcal{R}_{2} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\vert \\ &\quad\leq \sup_{r\leq R_{\varepsilon, \lambda}^{1}} 24 C_{1, \varepsilon }^{4\frac{\gamma +\xi -1}{\gamma} } \gamma ^{- \frac{1-\xi}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} \biggl(\frac{16}{(1+r^{2})^{4}} \biggr)^{ \frac{1 - \xi}{\gamma}} \\ &\qquad{}\times r^{4-\delta} e^{ \frac{\gamma + \xi - 1}{\gamma} G(\frac{\varepsilon x}{\tau _{1}}, x^{2}) + \frac{\gamma + \xi - 1}{\gamma \xi } G( \frac{\varepsilon x}{\tau _{1}}, x^{3})} \\ &\qquad{}\times \bigl\vert e^{\xi h_{2}^{1} + (1 - \xi ) h_{1}^{1}} - e^{ \xi k_{2}^{1} + (1 - \xi ) k_{1}^{1} } \bigr\vert \\ &\qquad{}+ \sup _{r \leq R_{ \varepsilon, \lambda}^{1}} r^{4-\delta} \biggl\vert \mathcal{L}_{ \lambda} \biggl({\frac{1}{\xi}}G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + h^{1}_{2}(x) \biggr) \\ &\qquad{}- \mathcal{L}_{\lambda} \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + k^{1}_{2}(x) \biggr) \biggr\vert \\ &\quad\leq c_{\kappa} \sup_{r\leq R_{\varepsilon, \lambda}^{1}} 24 C_{1, \varepsilon }^{4\frac{\gamma +\xi -1}{\gamma } } \gamma ^{- \frac{1-\xi}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} \biggl(\frac{16}{(1+r^{2})^{4}} \biggr)^{ \frac{1 - \xi}{ \gamma}}\\ &\qquad{}\times r^{4-\delta} \bigl[ \xi \bigl\vert h_{2}^{1} -k_{2}^{1} \bigr\vert + (1 - \xi ) \bigl\vert h_{1}^{1} - k_{1}^{1} \bigr\vert \bigr] \\ &\qquad{} +\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta} \biggl[ \bigl\vert \Delta \bigl(h^{1}_{2}-k^{1}_{2} \bigr) \bigr\vert \biggl(2 \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert \\ &\qquad{}+ {\frac{2}{\xi}} \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert + \bigl\vert \Delta h^{1}_{2} \bigr\vert + \bigl\vert \Delta k^{1}_{2} \bigr\vert \biggr) \\ &\qquad{} + \biggl\vert \nabla \bigl(h^{1}_{2}-k^{1}_{2} \bigr) \cdot \nabla \biggl(2 \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + {\frac{2}{\xi}}\Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + \Delta h^{1}_{2} + \Delta k^{1}_{2} \biggr)\\ &\qquad{} +\nabla \bigl(\Delta \bigl(h^{1}_{2}-k^{1}_{2} \bigr) \bigr) \\ & \qquad{}\times\nabla \biggl( 2 G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + { \frac{2}{\xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + h^{1}_{2} + k^{1}_{2} \biggr) \biggr\vert \biggr] +\lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \bigl\vert \Delta \bigl(h^{1}_{2}-k^{1}_{2} \bigr) \bigr\vert \\ &\qquad{}\times \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + h^{1}_{2}(x) \biggr) \biggr\vert ^{2} \\ &\qquad{}+ \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + k^{1}_{2}(x) \biggr) \biggr\vert ^{2} \biggr] \\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta} \biggl(2 \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert + {\frac{2}{\xi}} \biggl\vert \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert + \bigl\vert \Delta h^{1}_{2} \bigr\vert + \bigl\vert \Delta k^{1}_{2} \bigr\vert \biggr) \\ &\qquad{}\times \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + h^{1}_{2}(x) \biggr) \biggr\vert ^{2} \\ &\qquad{}- \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + k^{1}_{2}(x) \biggr) \biggr\vert ^{2} \biggr] \\ &\quad \leq c_{\kappa} \varepsilon ^{8\frac{\gamma +\xi -1}{\gamma}} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} \biggl( \frac{16}{(1+r^{2})^{4}} \biggr)^{\frac{1 - \xi}{ \gamma}} r^{4-\delta} \bigl[ \xi r^{\delta} \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4, \alpha}} + (1 - \xi )r^{\mu} \bigl\Vert h^{1}_{1}-k^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{ \mu}^{4,\alpha}} \bigr] \\ & \qquad{}+ {c}_{\kappa } \lambda \bigl( \varepsilon R_{ \varepsilon, \lambda} + \varepsilon ^{2} R_{\varepsilon, \lambda}^{2} +\varepsilon ^{3} R_{\varepsilon, \lambda}^{3} + R_{\varepsilon, \lambda}^{\delta } \bigl( \bigl\Vert h_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} + \bigl\Vert k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} \bigr) \bigr) \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} \\ &\qquad{}+ {c}_{\kappa} \lambda ^{2} \bigl( \varepsilon ^{2} R_{\varepsilon, \lambda}^{2} +\varepsilon ^{3} R_{\varepsilon, \lambda}^{3} \\ & \qquad{}+ \varepsilon R_{\varepsilon, \lambda}^{\delta + 1} \bigl( \bigl\Vert h_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} + \bigl\Vert k_{2}^{1} \bigr\Vert _{{ \mathcal {C}}_{\delta}^{4,\alpha}} \bigr) + \varepsilon ^{2}R_{ \varepsilon, \lambda}^{\delta + 2} \bigl( \bigl\Vert h_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{ \delta}^{4,\alpha}} + \bigl\Vert k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} \bigr)\\ &\qquad{} + R_{\varepsilon, \lambda}^{2\delta} \bigl( \bigl\Vert h_{2}^{1} \bigr\Vert _{{ \mathcal {C}}_{\delta}^{4,\alpha}} + \bigl\Vert k_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{ \delta}^{4,\alpha}} \bigr)^{2} \bigr) \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{ \mathcal {C}}_{\delta}^{4,\alpha}} \\ &\quad\leq c_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} + {c}_{\kappa }r_{\varepsilon, \lambda}^{2} \bigl\Vert h_{1}^{1}-k_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa } \lambda \bigl( r_{\varepsilon, \lambda} + \varepsilon ^{-\delta} r_{\varepsilon, \lambda}^{\delta +2} \bigr) \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} \\ &\qquad{}+ {c}_{\kappa} \lambda ^{2} \bigl(r_{\varepsilon, \lambda}^{2} +\varepsilon ^{-\delta} r_{\varepsilon, \lambda}^{ \delta +3} + \varepsilon ^{-2\delta} r_{\varepsilon, \lambda}^{2( \delta +2)} \bigr) \bigl\Vert h^{1}_{2}-k^{1}_{2} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4, \alpha}}. \end{aligned}$$
Using the following estimates
$$\begin{aligned} c_{\kappa }\varepsilon ^{-\delta} r_{\varepsilon, \lambda}^{2+\delta} \leq \textstyle\begin{cases} c_{\kappa }\varepsilon ^{1 - \frac{\delta}{2}} &\text{for } \varepsilon > \lambda, \\ c_{\kappa }\lambda ^{1 + \frac{\delta}{2}}\varepsilon ^{-\delta} &\text{for } \lambda > \varepsilon, \end{cases}\displaystyle \end{aligned}$$
together with condition \((A_{2})\), yield \({c}_{\kappa } \lambda ( r_{\varepsilon, \lambda} + \varepsilon ^{-\delta} r_{\varepsilon, \lambda}^{\delta +2} ) \leq {c}_{\kappa }r_{\varepsilon, \lambda}^{2}\) and \({c}_{\kappa }\lambda ^{2} (r_{\varepsilon, \lambda}^{2} + \varepsilon ^{-\delta} r_{\varepsilon, \lambda}^{\delta +3} + \varepsilon ^{-2\delta} r_{\varepsilon, \lambda}^{2(\delta +2)} )\leq {c}_{\kappa }r_{\varepsilon, \lambda}^{2}\). Making use of Proposition 1 together with (38) and using the condition \((A_{2})\) for \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma + \xi -1}{\xi}})\} )\), we get that there exists \(\bar{c}_{\kappa} > 0\) such that
$$\begin{aligned} \bigl\Vert \mathcal{ M}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr) -\mathcal{ M }_{1} \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}\leq \bar{c}_{ \kappa } r_{\varepsilon, \lambda}^{2} \bigl\Vert \bigl(h_{1}^{1},h_{2}^{1} \bigr)- \bigl(k_{1}^{1},k_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}. \end{aligned}$$
(53)
□
Reducing \(\varepsilon _{\kappa} \) and \(\lambda _{\kappa} \), if necessary, we can assume that \(\bar{c}_{\kappa} r_{\varepsilon, \lambda}^{2} < \frac{1}{2}\) for all \(\varepsilon \in (0, \varepsilon _{\kappa})\) and \(\lambda \in (0, \lambda _{\kappa})\). There exists also \(\gamma _{0} \in (0,1)\) such that \(\bar{c}_{\kappa} (1-\gamma ) \leq \frac{1}{2}\) for all \(\gamma \in (\gamma _{0},1)\). Therefore (52) and (53) are enough to show that
$$\begin{aligned} \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \mapsto \bigl( \mathcal{N}_{1} \bigl(h_{1}^{1}, h_{2}^{1} \bigr), \mathcal{M}_{1} \bigl(h_{1}^{1},h_{2}^{1} \bigr) \bigr) \end{aligned}$$
is a contraction from the ball
$$\begin{aligned} \bigl\{ \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \in {\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr) \times{\mathcal {C}}^{4, \alpha}_{\delta } \bigl( \mathbb{R}^{4} \bigr): \bigl\Vert \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2} \bigr\} \end{aligned}$$
into itself. Then, applying a contraction mapping argument, we obtain the following proposition.
Proposition 4
Given \(\kappa >0\), \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma +\xi -1}{\xi}})\} )\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\) and \(\gamma _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\), and for all \(\tau _{1}\) in some fixed compact subset of \([\tau _{1}^{-},\tau _{1}^{+}] \subset (0,\infty )\) there exists a unique \((h_{1}^{1}, h_{2}^{1}) (:= (h_{1,\varepsilon, \tau _{1}}, h_{2, \varepsilon, \tau _{1}, }))\) solution of (49) such that
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \bigr\Vert _{C^{4, \alpha}_{\mu}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\delta}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence (47) solves (44) in \({B_{R_{\varepsilon, \lambda}^{1}}}(x^{1}) \).
In \(B_{R_{\varepsilon }^{3}}(x^{3})\), following the same arguments as the first case by reversing the roles of the functions \(v_{1}\) and \(v_{2}\) and by respecting the changes of the coefficients, we can prove that there exists \((h_{1}^{3},h_{2}^{3})\in {\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\) such that
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{3}, h_{2}^{3} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
Furthermore, \((h_{1}^{3},h_{2}^{3})\) solves the equations
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} h_{1}^{3} = \frac{ 24 C_{3,\varepsilon }^{4 \frac{\gamma +\xi -1}{\xi }} 16^{\frac{1 - \gamma}{\xi} } \varepsilon ^{8 \frac {\gamma +\xi -1}{\xi}}}{\xi ^{\frac{1 - \gamma}{\xi}} (1+r^{2})^{ 4\frac{1 - \gamma}{\xi}}} e^{ \frac{ \gamma + \xi - 1}{\xi} G(\frac{\varepsilon x}{\tau _{3}}, x^{2}) + \frac{ \gamma + \xi - 1}{\gamma \xi} G( \frac{\varepsilon x}{\tau _{3}}, x^{1})+\gamma h_{1}^{3} + (1 - \gamma ) h_{2}^{3} } \\ \phantom{\Delta ^{2} h_{1}^{3} =}{} - \mathcal{L}_{\lambda} ({\frac{1}{\gamma}}G( \frac{\varepsilon x}{\tau _{3}}, x^{1}) +G( \frac{\varepsilon x}{\tau _{3}}, x^{2}) + h^{3}_{1}(x) ), \\ {\mathbb{L}} h_{2}^{3} = \frac{384 }{\xi (1 + r^{2})^{4}} [ e^{\xi h_{2}^{3} + (1 - \xi ) h_{1}^{3} } - \xi h_{2}^{3} - 1 ] - \mathcal{L}_{\lambda} ({\frac{1}{\xi}}\bar{u}(x-x^{3}) - {\frac{1-\xi}{\xi}} G( \frac{\varepsilon x}{\tau _{3}}, x^{2}) \\ \phantom{{\mathbb{L}} h_{2}^{3} = }{}- {\frac{1-\xi}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{3}}, x^{1})- \frac{\ln \xi}{\xi} + h^{3}_{2}(x) ). \end{cases}\displaystyle \end{aligned}$$
(54)
Then we have the following proposition.
Proposition 5
Given \(\kappa >0\), \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma +\xi -1}{\xi}})\} )\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\) and \(\xi _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\xi \in (\xi _{0}, 1)\) and for all \(\tau _{3}\) in some fixed compact subset of \([\tau _{3}^{-},\tau _{3}^{+}] \subset (0,\infty )\), there exists a unique \((h_{1}^{3}, h_{2}^{3}) (:= (h_{1,\varepsilon, \tau _{3}, }, h_{2, \varepsilon, \tau _{3}}))\) solution of (49) such that
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{3}, h_{2}^{3} \bigr) \bigr\Vert _{C^{4, \alpha}_{\delta}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\mu}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x):= \frac{1}{\gamma} G( \frac{\varepsilon x}{\tau _{3}}, x^{1})+ G( \frac{\varepsilon x}{\tau _{3}}, x^{2}) + h_{1}^{3}(x), \\ v_{2}(x) :=\frac{1}{\xi}\bar{u}(x-x^{3}) - \frac{1-\xi}{\xi} G(\frac{\varepsilon x}{\tau _{3}}, x^{2}) - \frac{1-\xi}{\gamma \xi} G(\frac{\varepsilon x}{\tau _{3}}, x^{1})- \frac{\ln \xi}{\xi} + h_{2}^{3}(x) \end{cases}\displaystyle \end{aligned}$$
solves (46) in \({B_{R_{\varepsilon, \lambda}^{3}}}(x^{3}) \).
In \(B_{R_{\varepsilon, \lambda}^{2}}(x^{2})\), we look for a solution of (45) of the form
$$\begin{aligned} \textstyle\begin{cases} v_{1} (x) = {\bar{u} }(x-x^{2}) + h_{1}^{2}(x), \\ v_{2} (x) = {\bar{u} }(x-x^{2}) + h_{2}^{2}(x). \end{cases}\displaystyle \end{aligned}$$
(55)
This amounts to solve the equations
$$\begin{aligned} \textstyle\begin{cases} {\mathbb{L}} h^{2}_{1} = \frac{384 }{(1 + r^{2})^{4}} [ e^{ \gamma h^{2}_{1} + (1 - \gamma ) h^{2}_{2} }-h^{2}_{1} - 1 ] - \mathcal{L}_{\lambda} ({\bar{u} }(x-x^{2}) + h_{1}^{2}(x) ), \\ {\mathbb{L}} h^{2}_{2} =\frac{384 }{(1 + r^{2})^{4}} [e^{ \xi h^{2}_{2} + (1 - \xi ) h^{2}_{1} } - h^{2}_{2} - 1 ] - \mathcal{L}_{\lambda} ( {\bar{u} }(x-x^{2}) + h_{2}^{2}(x) ). \end{cases}\displaystyle \end{aligned}$$
(56)
We denote by
$$\begin{aligned} {\mathbb{L}} h^{2}_{1} = \mathcal{R}_{3} \bigl(h_{1}^{2},h_{2}^{2} \bigr)\quad \text{and}\quad {\mathbb{L}} h^{2}_{2} = \mathcal{R}_{4} \bigl(h_{1}^{2},h_{2}^{2} \bigr). \end{aligned}$$
To find a solution of (56), it is enough to find a fixed point \((h_{1}^{2},h_{2}^{2})\) in a small ball of \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4})\times \mathcal{C}^{4, \alpha}_{\mu}(\mathbb{R}^{4})\), solutions of
$$\begin{aligned} \textstyle\begin{cases} h_{1}^{2} = \mathcal {G}_{\mu }\circ{\xi}_{{\mu}, R_{\varepsilon, \lambda}^{2}} \circ \mathcal{R}_{3}(h_{1}^{2},h_{2}^{2}) = \mathcal{N}_{2} (h_{1}^{2},h_{2}^{2}), \\ h_{2}^{2} = \mathcal {G}_{\mu }\circ{\xi}_{{\mu}, R_{\varepsilon, \lambda}^{2}} \circ \mathcal{R}_{4}(h_{1}^{2},h_{2}^{2}) = \mathcal{M}_{2} (h_{1}^{2},h_{2}^{2}). \end{cases}\displaystyle \end{aligned}$$
(57)
Then, we have the following result.
Lemma 2
Let \(\mu \in (1,2)\), \(\gamma _{0} \textit{ and } \xi _{0}\in (0,1)\). Given \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\) and \(\bar{c}_{\kappa} > 0\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0},1)\) and \(\xi \in (\xi _{0},1)\). We have
$$\begin{aligned} &\bigl\Vert \mathcal{ N}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})}\leq c_{\kappa } r_{\varepsilon, \lambda}^{2}, \qquad\bigl\Vert \mathcal{ M}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4} )}\leq c_{\kappa } r_{\varepsilon, \lambda}^{2},\\ &\bigl\Vert \mathcal{ N}_{2} \bigl(h_{1}^{2},h_{2}^{2} \bigr) -\mathcal{ N }_{2} \bigl(k_{1}^{2},k_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\\ &\quad\leq \bar{c}_{ \kappa } \bigl(1 - \gamma +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}\\ &\bigl\Vert \mathcal{M}_{2} \bigl(h_{1}^{2},h_{2}^{2} \bigr) -\mathcal{ M }_{2} \bigl(k_{1}^{2},k_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\\ &\quad\leq \bar{c}_{ \kappa }(1 - \xi ) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa} \bigl(1 - \xi +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}, \end{aligned}$$
provided \((h_{1}^{2}, h_{2}^{2})\), \((k_{1}^{2}, k_{2}^{2})\) in \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\) satisfying
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa }r_{\varepsilon, \lambda}^{2}\quad \textit{and}\quad \bigl\Vert \bigl(k_{1}^{2}, k_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{ \mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa }r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
(58)
Proof
We have
$$\begin{aligned} \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4- \mu} \bigl\vert \mathcal{R}_{3}(0,0) \bigr\vert \leq{}& \sup _{r \leq R_{ \varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \mathcal{L}_{\lambda} \bigl( \bar{u} \bigl(x-x^{2} \bigr) \bigr) \bigr\vert \\ \leq {}&\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl( \bigl\vert \Delta \bar{u} \bigl(x-x^{2} \bigr) \bigr\vert ^{2} + \bigl\vert \nabla \bar{u} \bigl(x-x^{2} \bigr) \nabla \bigl(\Delta \bar{u} \bigl(x-x^{2} \bigr) \bigr) \bigr\vert \bigr) \\ &{}+ \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \nabla \bar{u} \bigl(x-x^{2} \bigr) \bigr\vert ^{2} \bigl\vert \Delta \bar{u} \bigl(x-x^{2} \bigr) \bigr\vert . \end{aligned}$$
Making use of Proposition 1 together with (38), for \(\mu \in (1,2)\), we get that there exists \(c_{\kappa} > 0\) such that
$$\begin{aligned} \bigl\Vert \mathcal{N}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(59)
For the second estimate, we use the same techniques to prove
$$\begin{aligned} \bigl\Vert \mathcal{M}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(60)
To derive the third estimate, using the fact that for all functions in \({\mathcal {C}}_{\mu}^{k,\alpha}(\mathbb{R}^{4})\) bounded by a constant times \((1+r^{2})^{\mu /2}\) have their l-th partial derivatives that are bounded by \((1+r^{2})^{(\mu -l)/2}\), for \(l=1,\ldots,k+\alpha,\ldots \) (a.e. \(\vert \nabla ^{l}w \vert \leq c_{\kappa }r^{\mu - l} \Vert w\Vert _{{ \mathcal {C}}_{\mu}^{k,\alpha}(\mathbb{R}^{4})}\), \((1+r^{2})^{(\mu -l)/2} \sim r^{\mu -l}\) for r very large) and the fact that \(\vert \nabla ^{i} G(x,y) \vert \leq c\vert x-y\vert ^{-i}\) for \(i \geq 1\). Then for \((h_{1}^{2},h_{2}^{2}), (k_{1}^{2},k_{2}^{2})\) verifying (58), we get
$$\begin{aligned} & \sup_{r\leq R_{\varepsilon, \lambda}^{2}} r^{4 - \mu} \bigl\vert \mathcal{R}_{3} \bigl(h_{1}^{2},h_{2}^{2} \bigr) -\mathcal{R}_{3} \bigl(k_{1}^{2},k_{2}^{2} \bigr) \bigr\vert \\ &\quad\leq \sup_{r\leq R_{\varepsilon, \lambda}^{2}} \frac{384r^{4-\mu}}{(1+r^{2})^{4}} \bigl\vert \bigl( e^{\gamma h_{1}^{2} + (1 - \gamma ) h_{2}^{2} } - h_{1}^{2} \bigr) - \bigl( e^{\gamma k_{1}^{2} + (1 - \gamma ) k_{2}^{2} } - k_{1}^{2} \bigr) \bigr\vert \\ &\qquad{} + \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \mathcal{L}_{\lambda} \bigl(\bar{u} \bigl(x-x^{2} \bigr) +h^{2}_{1}(x) \bigr) - \mathcal{L}_{\lambda} \bigl( \bar{u} \bigl(x-x^{2} \bigr) +k^{2}_{1}(x) \bigr) \bigr\vert \\ & \quad\leq c \sup_{r\leq R_{\varepsilon, \lambda}^{2}} \frac{384 r^{4-\mu}}{(1+r^{2})^{4}} \bigl\vert (\gamma -1) \bigl(h_{1}^{2} -k_{1}^{2} \bigr) + (1 - \gamma ) \bigl(h_{2}^{2} - k_{2}^{2} \bigr) \bigr\vert +\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \Delta \bigl(h^{2}_{1}-k^{2}_{1} \bigr) \bigr\vert \\ &\qquad{}\times \bigl(2 \vert \Delta {\bar{u}} \vert + \bigl\vert \Delta h^{2}_{1} \bigr\vert + \bigl\vert \Delta k^{2}_{1} \bigr\vert \bigr) +\lambda \sup _{r \leq R_{ \varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \nabla \bigl(h^{2}_{1}-k^{2}_{1} \bigr) \cdot \nabla \bigl(2\Delta {\bar{u}} + \Delta h^{2}_{1} + \Delta k^{2}_{1} \bigr) \\ &\qquad{} +\nabla \bigl(\Delta \bigl(h^{2}_{1}-k^{2}_{1} \bigr) \bigr) \cdot \nabla \bigl(2 {\bar{u}} + h^{2}_{1} + k^{2}_{1} \bigr) \bigr\vert \\ &\qquad{}+\lambda ^{2} \sup _{r \leq R_{ \varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \Delta \bigl(h^{2}_{1}-k^{2}_{1} \bigr) \bigr\vert \bigl[ \bigl\vert \nabla \bigl(\bar{u} \bigl(x-x^{2} \bigr) +h^{2}_{1}(x) \bigr) \bigr\vert ^{2} \\ &\qquad{} + \bigl\vert \nabla \bigl(\bar{u} \bigl(x-x^{2} \bigr) +k^{2}_{1}(x) \bigr) \bigr\vert ^{2} \bigr]\\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \biggl( \frac{2}{\gamma} \vert \Delta {\bar{u}} \vert +2 \bigl\vert \Delta H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} \bigr\vert + \bigl\vert \Delta h^{2}_{1} \bigr\vert + \bigl\vert \Delta k^{2}_{1} \bigr\vert \biggr) \\ &\qquad{}\times \bigl[ \bigl\vert \nabla \bigl(\bar{u} \bigl(x-x^{2} \bigr) +h^{2}_{1}(x) \bigr) \bigr\vert ^{2} - \bigl\vert \nabla \bigl(\bar{u} \bigl(x-x^{2} \bigr) +k^{2}_{1}(x) \bigr) \bigr\vert ^{2} \bigr] \\ &\quad\leq c_{\kappa} (1 - \gamma ) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{ \mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa }\lambda \bigl( 1 + R_{\varepsilon, \lambda}^{\mu } \bigl( \bigl\Vert h_{1}^{2} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} \\ &\qquad{}+ \bigl\Vert k_{1}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \bigr) \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \\ &\qquad{} + {c}_{\kappa} \lambda ^{2} \bigl( 1 + R_{ \varepsilon, \lambda}^{\mu } \bigl( \bigl\Vert h_{1}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} + \bigl\Vert k_{1}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \bigr) + R_{\varepsilon, \lambda}^{2\mu} \bigl( \bigl\Vert h_{1}^{2} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + \bigl\Vert k_{1}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \bigr)^{2} \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \\ &\quad \leq c_{\kappa} (1 - \gamma ) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{ \mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa } \lambda \bigl( 1 + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \\ &\qquad{}\times + {c}_{\kappa} \lambda ^{2} \bigl( 1 + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} + \varepsilon ^{-2 \mu} r_{\varepsilon, \lambda}^{2(2+\mu )} \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}. \end{aligned}$$
Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get that there exists \(\bar{c}_{\kappa} > 0\) such that
$$\begin{aligned} &\bigl\Vert \mathcal{ N}_{2} \bigl(h_{1}^{2},h_{2}^{2} \bigr) -\mathcal{ N }_{2} \bigl(k_{1}^{2},k_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad\leq \bar{c}_{ \kappa } \bigl(1 - \gamma +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(61)
Similarly, we get
$$\begin{aligned} &\bigl\Vert \mathcal{M}_{2} \bigl(h_{1}^{2},h_{2}^{2} \bigr) -\mathcal{M}_{2} \bigl(k_{1}^{2},k_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad \leq \bar{c}_{ \kappa }(1 - \xi ) \bigl\Vert h^{2}_{1}-k^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa} \bigl(1 - \xi +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert h_{2}^{2}-k_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(62)
□
Then there exist \(\gamma _{0} \text{ and } \xi _{0} \in (0,1)\), reducing \(\varepsilon _{\kappa} \) and \(\lambda _{\kappa} \), if necessary, we can assume that \(\bar{c}_{\kappa } (1 - \gamma +r_{\varepsilon, \lambda}^{2})\leq 1/2\) and \(\bar{c}_{\kappa } (1 - \xi +r_{\varepsilon, \lambda}^{2})\leq 1/2\) for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\) and \(\xi \in (\xi _{0}, 1)\). Therefore (61) and (62) are enough to show that
$$\begin{aligned} \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \mapsto \bigl( \mathcal{N}_{2} \bigl(h_{1}^{2}, h_{2}^{2} \bigr), \mathcal{M}_{2} \bigl(h_{1}^{2},h_{2}^{2} \bigr) \bigr) \end{aligned}$$
is a contraction from the ball
$$\begin{aligned} \bigl\{ \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \in {\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr) \times{\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr): \bigl\Vert \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2} \bigr\} \end{aligned}$$
into itself. Then applying a contraction mapping argument, we obtain the following proposition.
Proposition 6
Given \(\kappa >0\), \(\mu \in (1, 2)\), \(\gamma _{0} \in (0,1)\) and \(\xi _{0} \in (0,1)\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\) and \(c_{\kappa} > 0\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\) and \(\xi \in (\xi _{0}, 1)\) and for all \(\tau _{2}\) in some fixed compact subset of \([\tau _{2}^{-},\tau _{2}^{+}] \subset (0,\infty )\), there exists a unique \((h_{1}^{2}, h_{2}^{2}) (:= (h_{1,\varepsilon, \tau _{2}}, h_{2, \varepsilon, \tau _{2}}))\) solution of (57) such that
$$\begin{aligned} \bigl\Vert \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \bigr\Vert _{C^{4, \alpha}_{\mu}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\mu}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence (55) solves (45) in \({B_{R_{\varepsilon, \lambda}^{2}}}(x^{2}) \).
3.1.3 Bi-harmonic extensions
Next, we will study the properties of interior and exterior bi-harmonic extensions. Given \((\varphi, \psi ), (\tilde{\varphi}, \tilde{\psi}) \in {\mathcal{C}}^{4, \alpha}(S^{3}) \times {\mathcal{C}}^{2,\alpha}(S^{3}) \), we define respectively \(H^{\mathrm{int}}=H^{\mathrm{int}} (\varphi, \psi; \cdot )=H^{\mathrm{int}}_{\varphi, \psi } \) and \(H^{\mathrm{ext}}=H^{\mathrm{ext}} (\tilde{\varphi}, \tilde{\psi}; \cdot )=H^{\mathrm{ext}}_{ \tilde{\varphi}, \tilde{\psi}} \) to be the solution of
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} H^{\mathrm{int}}= 0 & \text{in } B_{1}(0), \\ H^{\mathrm{int}} = \varphi & \text{on } \partial B_{1}(0), \\ \Delta H^{\mathrm{int}} = \psi& \text{on } \partial B_{1}(0), \end{cases}\displaystyle \end{aligned}$$
(63)
and
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} H^{\mathrm{ext}}= 0 & \text{in } \mathbb{R}^{4} - B_{1}(0), \\ H^{\mathrm{ext}} = \tilde{\varphi}&\text{on } \partial B_{1}(0), \\ \Delta H^{\mathrm{ext}} = \tilde{\psi}&\text{on } \partial B_{1}(0), \end{cases}\displaystyle \end{aligned}$$
(64)
which decays at infinity. We will also use
Definition 3
Given \(k\in {\mathbb{N}}\), \(\alpha \in (0,1)\) and \(\nu \in {\mathbb{R}}\), we define the space \({\mathcal{C}}^{k,\alpha}_{\nu }({\mathbb{R}}^{4} -B_{1}(0))\) as the space of functions \(w \in {\mathcal{C}}^{k,\alpha}_{\mathrm{loc}}({\mathbb{R}}^{4} -B_{1}(0))\) for which the following norm
$$\begin{aligned} \Vert w \Vert _{{\mathcal{C}}^{k,\alpha}_{\nu}({\mathbb{R}}^{4} -B_{1}(0))} = {\sup_{r\geq 1}} \bigl( r^{-\nu} \bigl\Vert w(r \cdot ) \bigr\Vert _{{\mathcal{C}}^{k,\alpha}_{\nu}(\bar{B}_{2}(0)-B_{1}(0))} \bigr) \end{aligned}$$
is finite.
We denote by \(e_{1}, \ldots, e_{4} \) the coordinate functions on \(S^{3}\).
Lemma 3
[4] Assume that
$$\begin{aligned} \int _{S^{3}} (8\varphi - \psi ) \,dv_{S^{3}} =0\quad \textit{and}\quad \int _{S^{3}} (12\varphi - \psi ) e_{\ell } \,dv_{S^{3}}=0\quad \textit{for } \ell = 1,\ldots,4. \end{aligned}$$
(65)
Then there exists \(c> 0\) such that
$$\begin{aligned} \bigl\Vert H^{\mathrm{int}}_{\varphi, \psi} \bigr\Vert _{{\mathcal{C}}_{2}^{ 4,\alpha}( \bar{B}_{1}^{*}(0))} \leq c \bigl( \Vert {\varphi} \Vert _{{\mathcal{C}}^{ 4,\alpha}( S^{3})}+ \Vert {\psi} \Vert _{{\mathcal{C}}^{2,\alpha}( S^{3})} \bigr). \end{aligned}$$
Similarly, there exists \(c> 0\) such that if
$$\begin{aligned} \int _{S^{3}} \tilde{\psi}\,dv_{S^{3}} = 0, \end{aligned}$$
(66)
then
$$\begin{aligned} \bigl\Vert H^{\mathrm{ext}}_{\tilde{\varphi}, \tilde{\psi}} \bigr\Vert _{{\mathcal{C}}_{-1}^{ 4, \alpha} ({\mathbb{R}}^{4} -B_{1}(0))}\leq c \bigl( \Vert {\tilde{\varphi}} \Vert _{{\mathcal{C}}^{4,\alpha}( S^{3})}+ \Vert {\tilde{\psi}} \Vert _{{\mathcal{C}}^{2, \alpha}( S^{3})} \bigr). \end{aligned}$$
If \(F \subset L^{2}(S^{3})\) is a subspace \(S^{3}\), we denote \(F^{\perp}\) to be the subspace of F, which are \(L^{2}(S^{3})\)-orthogonal to the functions \(1,e_{1},\ldots,e_{4}\). We will need the following result.
Lemma 4
[4] The mapping
$$\begin{aligned} & \mathcal{P}: \mathcal{C}^{4,\alpha} \bigl(S^{3} \bigr)^{\perp }\times \mathcal{C}^{2,\alpha} \bigl(S^{3} \bigr)^{\perp } \longrightarrow \mathcal{C}^{3, \alpha} \bigl(S^{3} \bigr)^{\perp }\times \mathcal{C}^{1,\alpha} \bigl(S^{3} \bigr)^{\perp }, \\ &(\varphi, \psi ) \longmapsto \bigl(\partial _{r} \bigl(H^{\mathrm{int}}_{ \varphi, \psi} - H^{\mathrm{ext}}_{\varphi, \psi} \bigr), \partial _{r} \bigl(\Delta H^{\mathrm{int}}_{ \varphi, \psi} - \Delta H^{\mathrm{ext}}_{\varphi, \psi} \bigr) \bigr) \end{aligned}$$
is an isomorphism.
3.2 The nonlinear interior problem
Here, we are looking for a solution of the following systems as in the above subsection, we only add the interior harmonic extension and the perturbation term \(v_{i}^{j}\) for \(i,j = 1, 2\).
$$\begin{aligned} &\textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1})= 24 e^{\gamma v_{1} + (1-\gamma ) v_{2}} &\text{in } B_{R_{\varepsilon, \lambda}^{1}}(x^{1}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2}) = 24 C_{1,\varepsilon }^{4\frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} e^{\xi v_{2} +(1-\xi ) v_{1}}& \text{in } B_{R_{\varepsilon, \lambda}^{1}}(x^{1}), \end{cases}\displaystyle \end{aligned}$$
(67)
$$\begin{aligned} &\textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1}) = 24 e^{\gamma v_{1} + (1-\gamma ) v_{2}} & \text{in } B_{R_{\varepsilon, \lambda}^{2}}(x^{2}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2})= 24 e^{\xi v_{2} +(1- \xi ) v_{1}} &\text{in } B_{R_{\varepsilon, \lambda}^{2}}(x^{2}) \end{cases}\displaystyle \end{aligned}$$
(68)
and
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} v_{1} + \mathcal{L}_{\lambda}(v_{1})= 24 C_{3, \varepsilon }^{4\frac{\gamma +\xi -1}{\xi}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\xi}} e^{\gamma v_{1} + (1-\gamma ) v_{2}} & \text{in } B_{R_{\varepsilon, \lambda}^{3}}(x^{3}), \\ \Delta ^{2} v_{2} + \mathcal{L}_{\lambda}(v_{2})= 24 e^{\xi v_{2} +(1-\xi ) v_{1}} & \text{in } B_{R_{\varepsilon, \lambda}^{3}}(x^{3}), \end{cases}\displaystyle \end{aligned}$$
(69)
where \(\mathcal{L}_{\lambda}\) is defined by (40) and \(C_{i,\varepsilon }={\frac{2}{\tau _{i} (1+\varepsilon ^{2})}}\) for \(i=1,3\). Here \(\tau _{i} > 0\) is a constant, which will be fixed later.
Given \(\varphi ^{i}:=(\varphi _{1}^{i}, \varphi _{2}^{i}) \in ({\mathcal {C}}^{4, \alpha}(S^{3}))^{2}\) and \(\psi ^{i}:=(\psi _{1}^{i}, \psi _{2}^{i}) \in ({\mathcal {C}}^{2, \alpha}(S^{3}))^{2}\) such that \((\varphi _{1}^{i}, \psi _{1}^{i})\) and \((\varphi _{2}^{i}, \psi _{2}^{i})\) are satisfying (65). We denote by \(\bar{u}=u_{\varepsilon =1,\tau _{i}=1}\), we write for \(x \in B_{R^{1}_{\varepsilon, \lambda}}(x^{1})\) the following system
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x) = {\frac{1}{\gamma}}\bar{u}(x-x^{1}) -{\frac{1-\gamma}{\gamma}} G( \frac{\varepsilon x}{\tau _{1}},x^{2})- {\frac{1-\gamma}{\gamma \xi}} G( \frac{\varepsilon x}{\tau _{1}},x^{3})- \frac{\ln \gamma}{\gamma} + h^{1}_{1}(x) \\ \phantom{v_{1}(x) =}{}+ H^{\mathrm{int}}(\varphi ^{1}_{1}, \psi ^{1}_{1}; {\frac{x-x^{1}}{R_{ \varepsilon, \lambda}^{1}}})+ v^{1}_{1}(x), \\ v_{2}(x) = {\frac{1}{\xi}}G(\frac{\varepsilon x}{\tau _{1}}, x^{3}) +G( \frac{\varepsilon x}{\tau _{1}}, x^{2}) + h^{1}_{2}(x) + H^{\mathrm{int}}( \varphi ^{1}_{2}, \psi ^{1}_{2};{\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}}) + v^{1}_{2}(x). \end{cases}\displaystyle \end{aligned}$$
Using the fact that \(H^{\mathrm{int}}\) is bi-harmonic and that \(e^{\bar{u}(x-x^{1})}=\frac{16}{(1+|x-x^{1}|^{2})^{4}}\), we see that this amounts to solve the system
$$\begin{aligned} \textstyle\begin{cases} {\mathbb{L}} v^{1}_{1} = { \frac{384 }{\gamma (1 + r^{2})^{4}} (e^{\gamma (h^{1}_{1}+ H^{\mathrm{int}}_{ \varphi _{1}^{1}, \psi _{1}^{1} }+v^{1}_{1} )+(1-\gamma ) (h^{1}_{2}+H^{\mathrm{int}}_{ \varphi _{2}^{1}, \psi _{2}^{1} }+v^{1}_{2} )} - \gamma v^{1}_{1} -1 )}- \mathcal{L}_{\lambda} ({\frac{1}{\gamma}}\bar{u}(x-x^{1}) \\ \phantom{{\mathbb{L}} v^{1}_{1} =}{} - {\frac{1-\gamma}{\gamma}} G(\frac{\varepsilon x}{\tau _{1}},x^{2})- {\frac{1- \gamma}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})- \frac{\ln \gamma}{\gamma} \\ \phantom{{\mathbb{L}} v^{1}_{1} =}{}+H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} ({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}})+ h^{1}_{1}(x)+ v^{1}_{1}(x) ) - \Delta ^{2} h_{1}^{1}, \\ \Delta ^{2} v^{1}_{2} = \frac{24 C_{1,\varepsilon }^{4\frac{\gamma +\xi -1}{\gamma }} 16^{\frac{1-\xi}{\gamma}} \varepsilon ^{8\frac{\gamma +\xi -1}{\gamma}}}{\gamma ^{\frac{1-\xi}{\gamma}}(1+r^{2})^{4{\frac{1-\xi}{\gamma}}}} \\ \phantom{\Delta ^{2} v^{1}_{2} =}{}\times e^{{\frac{\gamma +\xi -1}{\gamma}}G(\frac{\varepsilon x}{\tau _{1}}, x^{2})+{ \frac{\gamma + \xi -1}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})+ \xi ( h^{1}_{2}+H^{\mathrm{int}}_{\varphi _{2}^{1}, \psi _{2}^{1} }+v^{1}_{2} ) +(1-\xi ) (h^{1}_{1}+ H^{\mathrm{int}}_{\varphi _{1}^{1}, \psi _{1}^{1} } +v^{1}_{2} )} \\ \phantom{\Delta ^{2} v^{1}_{2} =}{}- \mathcal{L}_{\lambda} ({\frac{1}{\xi}}G( \frac{\varepsilon x}{\tau _{1}}, x^{3}) +G( \frac{\varepsilon x}{\tau _{1}}, x^{2}) + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}}({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}}) + h^{1}_{2}(x)+ v^{1}_{2}(x) ) -\Delta ^{2} h_{2}^{1}, \end{cases}\displaystyle \end{aligned}$$
(70)
We denote by
$$\begin{aligned} {\mathbb{L}} v^{1}_{1} = \mathscr{R}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr)\quad \text{and}\quad \Delta ^{2} v^{1}_{2} = \mathscr{R}_{2} \bigl(v_{1}^{1},v_{2}^{1} \bigr). \end{aligned}$$
Fix \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma + \xi -1}{\xi}})\} )\). To find a solution of (48), it is enough to find a fixed point \((v_{1}^{1},v_{2}^{1})\) in a small ball of \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4}) \times \mathcal{C}^{4, \alpha}_{\delta}(\mathbb{R}^{4})\) solutions of
$$\begin{aligned} \textstyle\begin{cases} v_{1}^{1} = \mathcal {G}_{\mu }\circ{\xi}_{\mu, R_{\varepsilon, \lambda}^{1}} \circ \mathscr{R}_{1}(v_{1}^{1},v_{2}^{1}) = \mathscr{N}_{1}(v_{1}^{1},v_{2}^{1}), \\ v_{2}^{1} = {\mathcal {K}}_{\delta }\circ{\xi}_{\delta, R_{ \varepsilon,\lambda}^{1}} \circ \mathscr{R}_{2}(v_{1}^{1},v_{2}^{1})= \mathscr{M}_{1}(v_{1}^{1},v_{2}^{1}). \end{cases}\displaystyle \end{aligned}$$
(71)
Here \(\xi _{\mu,R_{\varepsilon,\lambda}^{1}}\) is defined in (37), \({\mathcal {G}}_{\mu}\) and \({\mathcal {K}}_{\delta}\) are defined after Propositions 1, 2, respectively.
Given \(\kappa >0\) (whose value will be fixed later), we further assume that the functions \(\varphi _{j}^{1}\) and \(\psi _{j}^{1}\) satisfy
$$\begin{aligned} \bigl\Vert \varphi _{j}^{1} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} \leq \kappa r_{ \varepsilon, \lambda}^{2} \quad\text{and}\quad \bigl\Vert \psi _{j}^{1} \bigr\Vert _{{ \mathcal {C}}^{2, \alpha}(S^{3})} \leq \kappa r_{\varepsilon, \lambda}^{2}, \quad\text{for } j=1, 2. \end{aligned}$$
(72)
Then we have the following result.
Lemma 5
Let \(\varphi ^{1}:=(\varphi _{1}^{1}, \varphi _{2}^{1}) \in ({\mathcal {C}}^{4, \alpha}(S^{3}))^{2}\) and \(\psi ^{1}:=(\psi _{1}^{1}, \psi _{2}^{1}) \in ({\mathcal {C}}^{2, \alpha}(S^{3}))^{2}\) such that \((\varphi _{1}^{1}, \psi _{1}^{1})\) and \((\varphi _{2}^{1}, \psi _{2}^{1})\) are satisfying (65) and (72). Given \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0 \), \(\bar{c}_{\kappa} > 0 \) and \(\gamma _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0},1)\), \(\mu \in (1,2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma + \xi -1}{\xi}})\} )\). We have
$$\begin{aligned} &\bigl\Vert \mathcalligra{ }\mathscr{N}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}, \qquad\bigl\Vert \mathcalligra{ }\mathscr{M}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4} )} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2},\\ &\bigl\Vert \mathcalligra{ }\mathscr{N}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr) -\mathcalligra{ }\mathscr{N}\mathcalligra{ }_{1} \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\leq \bar{c}_{ \kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert v^{1}_{1}-t^{1}_{1} \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} +\bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{1}-t_{2}^{1} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4} )} \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert \mathcalligra{ }\mathscr{M}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr) -\mathcalligra{ }\mathscr{M}\mathcalligra{ }_{1} \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq \bar{c}_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert \bigl(v_{1}^{1},v_{2}^{1} \bigr)- \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}, \end{aligned}$$
provided \((v_{1}^{1}, v_{2}^{1}), (t_{1}^{1}, t_{2}^{1}) \in {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{ \delta }(\mathbb{R}^{4})\) satisfying
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{1}, v_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2}, \qquad\bigl\Vert \bigl(t_{1}^{1} , t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4}) \times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{ \kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(73)
Proof
The proof of the first and the second estimates follows from the asymptotic behavior of \(H^{\mathrm{int}}\) together with the assumption on the norms of \(\varphi ^{1}_{j}\) and \(\psi ^{1}_{j}\) given by (72) and it follows from the estimate of \(H^{\mathrm{int}}\), given by Lemma 3, that
$$\begin{aligned} \biggl\Vert H^{\mathrm{int}}_{\varphi ^{1}_{j}, \psi ^{1}_{j}} \biggl({\frac{r}{R^{1}_{ \varepsilon,\lambda}}} \cdot \biggr) \biggr\Vert _{{\mathcal {C}}^{4, \alpha} ( \bar{B}_{2}(0) - B_{1}(0))} \leq C r^{2} \bigl(R^{1}_{\varepsilon,\lambda} \bigr)^{-2} \bigl( \bigl\Vert \varphi ^{1}_{j} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} + \bigl\Vert \psi ^{1}_{j} \bigr\Vert _{{\mathcal {C}}^{2, \alpha}(S^{3})} \bigr), \end{aligned}$$
for all \(r \leq {\frac{R^{1}_{\varepsilon,\lambda}}{2}}\). Then by (72), we get
$$\begin{aligned} \biggl\Vert H^{\mathrm{int}}_{\varphi ^{1}_{j}, \psi ^{1}_{j}} \biggl({ \frac{r}{R^{1}_{ \varepsilon, \lambda}}}\cdot \biggr) \biggr\Vert _{{\mathcal {C}}^{4, \alpha} ( \bar{B}_{2}(0) - B_{1}(0))} \leq c_{\kappa }\varepsilon ^{2} r^{2}. \end{aligned}$$
(74)
On the other hand,
$$\begin{aligned} &\sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \mathscr{R}_{1}(0,0) \bigr\vert \\ &\quad\leq \sup_{r \leq R_{\varepsilon,\lambda}^{1}} \frac{384 r^{4-\mu} }{(1 + r^{2})^{4}} \frac{1}{\gamma} \bigl\vert e^{ \gamma (H^{\mathrm{int}}_{\varphi _{1}^{1}, \psi _{1}^{1} }+h^{1}_{1} ) + (1 - \gamma ) ( H^{\mathrm{int}}_{\varphi _{2}^{1}, \psi _{2}^{1} }+h^{1}_{2} )} - 1 \bigr\vert + \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \biggl( \biggl\vert \mathcal{L}_{\lambda} \biggl({ \frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) \\ &\qquad{} - {\frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} \\ &\qquad{}+ H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr)+ h^{1}_{1}(x) \biggr) \biggr\vert + \bigl\vert \Delta ^{2}h^{1}_{1} \bigr\vert \biggr) \\ &\quad\leq \sup_{r \leq R_{\varepsilon, \lambda}^{1}} \frac{ 384 r^{4-\mu} }{(1+r^{2})^{4}} \frac{1}{\gamma} \bigl( \gamma r^{2} \bigl\Vert H^{\mathrm{int}}_{\varphi _{1}^{1}, \psi _{1}^{1} } \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{2}}+ \gamma r^{\mu} \bigl\Vert h^{1}_{1} \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{\mu }} \\ &\qquad{}+ (1 - \gamma ) r^{2} \bigl\Vert H^{\mathrm{int}}_{ \varphi _{2}^{1}, \psi _{2}^{1} } \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{2}} + (1 - \gamma ) r^{\mu } \bigl\Vert h^{1}_{2} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{ \mu}} \bigr) \\ &\qquad{} +\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \biggl( \Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - { \frac{1- \gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr)\\ &\qquad{} - { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) + H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}} \biggr) + h^{1}_{1}(x) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + 2\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - { \frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) + H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}} \biggr) + h^{1}_{1}(x) \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \nabla \biggl(\Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1- \gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \\ &\qquad{}+ H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}} \biggr)+ h^{1}_{1}(x) \biggr) \biggr) \biggr\vert \\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \\ &\qquad{}+ H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({\frac{x-x^{1}}{R_{ \varepsilon,\lambda}^{1}}} \biggr) \biggr)+ h^{1}_{1}(x) \biggr\vert ^{2} \\ &\qquad{}\times \biggl\vert \Delta \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}},x^{3} \biggr) \\ &\qquad{}+ H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({\frac{x-x^{1}}{R_{\varepsilon, \lambda}^{1}}} \biggr) + h^{1}_{1}(x) \biggr) \biggr\vert +\sup _{r \leq R_{ \varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \Delta ^{2}h^{1}_{1} \bigr\vert \\ &\quad\leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{\kappa }\lambda \bigl(1 + \varepsilon ^{4} R_{\varepsilon, \lambda}^{4-\mu}+\varepsilon ^{2} R_{ \varepsilon, \lambda}^{2-\mu} + \varepsilon ^{3} R_{\varepsilon, \lambda}^{3-\mu} + \varepsilon ^{6} R_{\varepsilon, \lambda}^{5-\mu} + \varepsilon \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}\\ &\qquad{} + \varepsilon ^{2} R_{\varepsilon, \lambda}^{2} \bigl\Vert h_{1}^{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + \varepsilon ^{3} R_{\varepsilon, \lambda}^{3} \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} \\ & \qquad{}+ R_{\varepsilon, \lambda}^{\mu } \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}}^{2} \bigr) + c_{\kappa }\lambda ^{2} \bigl(1 + \varepsilon R_{ \varepsilon, \lambda}^{2-\mu} + \varepsilon ^{2} R_{\varepsilon, \lambda}^{3-\mu} + \varepsilon ^{3} R_{\varepsilon, \lambda}^{4-\mu} + \varepsilon ^{5} R_{\varepsilon, \lambda}^{5-\mu} \\ &\qquad{}+ R_{\varepsilon, \lambda} \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + \varepsilon ^{2} R_{\varepsilon, \lambda}^{3} \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \\ &\qquad{} + \varepsilon R_{\varepsilon, \lambda}^{2} \bigl\Vert h_{1}^{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} + R_{\varepsilon, \lambda}^{\mu +1} \bigl\Vert h_{1}^{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}^{2} \bigr) \\ &\quad \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{\kappa } \lambda \bigl(1 + \varepsilon r_{\varepsilon, \lambda}^{2} + \varepsilon ^{\mu }r_{ \varepsilon, \lambda}^{2-\mu} + \varepsilon ^{-\mu}r_{\varepsilon, \lambda}^{\mu +4} \bigr)\\ &\qquad{} + c_{\kappa } \lambda ^{2} \bigl(1 + \varepsilon ^{\mu -1} r_{\varepsilon, \lambda}^{2-\mu} + \varepsilon ^{\mu} r_{ \varepsilon, \lambda}^{5-\mu} + \varepsilon ^{-1} r_{\varepsilon, \lambda}^{3} + \varepsilon ^{-\mu -1} r_{\varepsilon, \lambda}^{\mu +5} \bigr). \end{aligned}$$
Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get that there exists \(c_{\kappa} > 0\) such that
$$\begin{aligned} \bigl\Vert \mathscr{N}_{1}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(75)
For the second estimate, we have
$$\begin{aligned} &\sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \bigl\vert \mathscr{R}_{2}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa }\sup _{r \leq R_{ \varepsilon, \lambda}^{1}} C_{1,\varepsilon }^{4 \frac{\gamma +\xi -1}{\gamma}} \varepsilon ^{8 \frac{\gamma +\xi -1}{\gamma}} r^{4-\delta} \biggl( \frac{16}{(1+r^{2})^{4}} \biggr)^{ \frac{1 - \xi}{\gamma} } \\ &\qquad{}\times e^{\frac{\gamma + \xi - 1}{\gamma} G( \frac{\varepsilon x}{\tau _{1}}, x^{2}) + \frac{\gamma + \xi - 1}{\gamma \xi } G( \frac{\varepsilon x}{\tau _{1}}, x^{3})} e^{ \xi (H^{\mathrm{int}}_{ \varphi _{2}^{1}, \psi _{2}^{1} }+ h^{1}_{2} ) + (1 - \xi ) ( H^{\mathrm{int}}_{ \varphi _{1}^{1}, \psi _{1}^{1} }+ h^{1}_{1} )} \\ &\qquad{} + \sup_{r\leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \biggl( \biggl\vert \mathcal{L}_{\lambda} \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr)\\ &\qquad{} + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr) + h^{1}_{2}(x) \biggr) \biggr\vert + \bigl\vert \Delta ^{2}h^{1}_{2} \bigr\vert \biggr) \\ &\quad \leq c_{\kappa }\sup_{r \leq R_{\varepsilon, \lambda}^{1}} \varepsilon ^{8\frac{\gamma + \xi -1}{\gamma }} r^{4- \delta} \biggl( \frac{16}{(1+r^{2})^{4}} \biggr)^{ \frac{1 - \xi}{\gamma} } \bigl(\xi r^{2} \bigl\Vert H^{\mathrm{int}}_{\varphi _{2}^{1}, \psi _{2}^{1} } \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{2}} + r^{\mu } \bigl\Vert h^{1}_{2} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }}\\ &\qquad{} + (1 - \xi ) r^{2} \bigl\Vert H^{\mathrm{int}}_{ \varphi _{1}^{1}, \psi _{1}^{1} } \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{2}} \\ &\qquad{} + (1 - \xi ) r^{\mu} \bigl\Vert h^{1}_{1} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{ \mu }} + 1 \bigr) +\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \biggl\vert \Delta \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}+ H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr)+ h^{1}_{2}(x) \biggr) \biggr\vert ^{2} \\ &\qquad{} + \lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \delta} \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({\frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr)+ h^{1}_{2}(x) \biggr) \biggr\vert \\ & \qquad{}\times\biggl\vert \nabla \biggl(\Delta \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr)+ h^{1}_{2}(x) \biggr) \biggr) \biggr\vert \\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\delta} \biggl\vert \nabla \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \\ &\qquad{} +G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({\frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr)+ h^{1}_{2}(x) \biggr) \biggr\vert ^{2} \biggl\vert \Delta \biggl({\frac{1}{\xi}}G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \\ &\qquad{}+G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{2}, \psi ^{1}_{2}} \biggl({\frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr) + h^{1}_{2}(x) \biggr) \biggr\vert \\ &\qquad{} +\sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \Delta ^{2}h^{1}_{2} \bigr\vert . \end{aligned}$$
With the same argument as above, but using the condition \((A_{2})\), we get \(\|\mathscr{M}_{1}(0,0) \|_{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})}\leq c_{\kappa }r_{\varepsilon, \lambda}^{2}\).
To derive the third estimate, for \((v_{1}^{1},v_{2}^{1}), (t_{1}^{1},t_{2}^{1})\) verifying (73), we have
$$\begin{aligned} &\sup_{r\leq R_{\varepsilon, \lambda}^{1}} r^{4 - \mu} \bigl\vert \mathscr{R}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr)-\mathscr{R}_{1} \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\vert \\ &\quad\leq \sup_{r\leq R_{\varepsilon, \lambda}^{1}} \frac{384r^{4-\mu}}{(1+r^{2})^{4}} \frac{1}{\gamma} \bigl\vert \bigl( e^{ \gamma (h^{1}_{1}+ H^{\mathrm{int}}_{\varphi _{1}^{1}, \psi _{1}^{1} }+v^{1}_{1} )+(1-\gamma ) (h^{1}_{2}+H^{\mathrm{int}}_{\varphi _{2}^{1}, \psi _{2}^{1} }+v^{1}_{2} )} - \gamma v_{1}^{1} \bigr) \\ &\qquad{} - \bigl( e^{\gamma (h^{1}_{1}+ H^{\mathrm{int}}_{\varphi _{1}^{1}, \psi _{1}^{1} }+t^{1}_{1} )+(1-\gamma ) (h^{1}_{2}+H^{\mathrm{int}}_{\varphi _{2}^{1}, \psi _{2}^{1} }+t^{1}_{2} )} - \gamma t_{1}^{1} \bigr) \bigr\vert \\ &\qquad{} + \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \mathcal{L}_{\lambda} \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ & \qquad{}- {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} + H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr) +h^{1}_{1}(x)+v^{1}_{1}(x) \biggr) \\ &\qquad{}- \mathcal{L}_{\lambda} \biggl({\frac{1}{\gamma}}\bar{u} \bigl(x-x^{1} \bigr) - {\frac{1- \gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- {\frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr)- \frac{\ln \gamma}{\gamma} + H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} \biggl({ \frac{x-x^{1}}{R_{\varepsilon,\lambda}^{1}}} \biggr) +h^{1}_{1}(x)+t^{1}_{1}(x) \biggr) \biggr\vert \\ &\quad\leq c_{\kappa} \sup_{r\leq R_{\varepsilon, \lambda}^{1}} \frac{384 r^{4-\mu}}{(1+r^{2})^{4}} \frac{1}{\gamma} \bigl[ \gamma ^{2} \bigl( \bigl(v_{1}^{1} \bigr)^{2}- \bigl(t_{1}^{1} \bigr)^{2} \bigr) + (1 - \gamma ) \bigl\vert v_{2}^{1} - t_{2}^{1} \bigr\vert \bigr] +\lambda \sup _{r\leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \bigl\vert \Delta \bigl(v^{1}_{1}-t^{1}_{1} \bigr) \bigr\vert \\ &\qquad{}\times \biggl(\frac{2}{\gamma} \vert \Delta {\bar{u}} \vert + 2{ \frac{1-\gamma}{\gamma}} \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert + 2{\frac{1- \gamma}{\gamma \xi}} \biggl\vert \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert +2 \bigl\vert \Delta H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \bigr\vert \\ &\qquad{}+ 2 \bigl\vert \Delta h^{1}_{1} \bigr\vert + \bigl\vert \Delta v^{1}_{1} \bigr\vert + \bigl\vert \Delta t^{1}_{1} \bigr\vert \biggr) \\ &\qquad{}+2 \lambda \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \biggl\vert \nabla \bigl(v^{1}_{1}-t^{1}_{1} \bigr) \cdot \nabla \biggl( \frac{2}{\gamma} \Delta {\bar{u}} + 2{\frac{1-\gamma}{\gamma}} \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}+ 2{ \frac{1-\gamma}{\gamma \xi}} \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +2 \Delta H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} + 2\Delta h^{1}_{1} \\ & \qquad{}+ \Delta v^{1}_{1} + \Delta t^{1}_{1} \biggr) +\nabla \bigl(\Delta \bigl(v^{1}_{1}-t^{1}_{1} \bigr) \bigr) \cdot \nabla \biggl(\frac{2}{\gamma} {\bar{u}} - 2{ \frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- 2{ \frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) +2 H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} + 2h^{1}_{1} \\ &\qquad{} + v^{1}_{1} + t^{1}_{1} \biggr) \biggr\vert +\lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \Delta \bigl(v^{1}_{1}-t^{1}_{1} \bigr) \bigr\vert \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\gamma}} \bar{u}\\ &\qquad{} - {\frac{1- \gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - {\frac{1- \gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} \\ & \qquad{}+h^{1}_{1} +v^{1}_{1} \biggr) \biggr\vert ^{2} + \biggl\vert \nabla \biggl({\frac{1}{\gamma}} \bar{u} - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \\ &\qquad{}- {\frac{1-\gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + H^{\mathrm{int}}_{\varphi ^{1}_{1}, \psi ^{1}_{1}} +h^{1}_{1} +t^{1}_{1} \biggr) \biggr\vert ^{2} \biggr] \\ & \qquad{}+ \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{1}} r^{4- \mu} \biggl(\frac{2}{\gamma} \vert \Delta {\bar{u}} \vert + 2{ \frac{1-\gamma}{\gamma}} \biggl\vert \Delta G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) \biggr\vert + 2{\frac{1-\gamma}{\gamma \xi}} \biggl\vert \Delta G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) \biggr\vert \\ &\qquad{}+2 \bigl\vert \Delta H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} \bigr\vert + 2 \bigl\vert \Delta h^{1}_{1} \bigr\vert \\ &\qquad{} + \bigl\vert \Delta v^{1}_{1} \bigr\vert + \bigl\vert \Delta t^{1}_{1} \bigr\vert \biggr) \biggl[ \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} - {\frac{1-\gamma}{\gamma}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr)\\ &\qquad{} - {\frac{1-\gamma}{\gamma \xi}} G \biggl( \frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} +h^{1}_{1} +v^{1}_{1} \biggr) \biggr\vert ^{2} \\ &\qquad{}- \biggl\vert \nabla \biggl({\frac{1}{\gamma}}\bar{u} - { \frac{1-\gamma}{\gamma}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{2} \biggr) - { \frac{1-\gamma}{\gamma \xi}} G \biggl(\frac{\varepsilon x}{\tau _{1}}, x^{3} \biggr) + H^{\mathrm{int}}_{ \varphi ^{1}_{1}, \psi ^{1}_{1}} +h^{1}_{1}+t^{1}_{1} \biggr) \biggr\vert ^{2} \biggr] \\ & \quad\leq c_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert v^{1}_{1}-t^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{1}-t_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}} + {c}_{\kappa } \lambda \bigl( 1 + r_{\varepsilon, \lambda} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{\mu +2} \bigr) \bigl\Vert v^{1}_{1}-t^{1}_{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}} \\ & \qquad{}+ {c}_{\kappa} \lambda ^{2} \bigl( 1 + r_{ \varepsilon, \lambda} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{ \mu +2} + \varepsilon ^{-2\mu} r_{\varepsilon, \lambda}^{2\mu +4} \bigr) \bigl\Vert v^{1}_{1}-t^{1}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}. \end{aligned}$$
Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get that there exists \(\bar{c}_{\kappa} > 0\) such that
$$\begin{aligned} &\bigl\Vert \mathcalligra{ }\mathscr{N}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr) -\mathcalligra{ }\mathscr{N}\mathcalligra{ }_{1} \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad\leq \bar{c}_{ \kappa } r_{\varepsilon, \lambda}^{2} \bigl\Vert v^{1}_{1}-t^{1}_{1} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{1}-t_{2}^{1} \bigr\Vert _{{\mathcal {C}}_{\delta}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(76)
Similarly, we get the estimate for
$$\begin{aligned} \bigl\Vert \mathcalligra{ }\mathscr{M}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr) -\mathcalligra{ }\mathscr{M}\mathcalligra{ }_{1} \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}\leq \bar{c}_{ \kappa } r_{\varepsilon, \lambda}^{2} \bigl\Vert \bigl(v_{1}^{1},v_{2}^{1} \bigr)- \bigl(t_{1}^{1},t_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})}. \end{aligned}$$
(77)
□
Reducing \(\varepsilon _{\kappa} \) and \(\lambda _{\kappa} \), if necessary, we can assume that \(\bar{c}_{\kappa} r_{\varepsilon, \lambda}^{2} < \frac{1}{2}\) for all \(\varepsilon \in (0, \varepsilon _{\kappa})\) and \(\lambda \in (0, \lambda _{\kappa})\). There exists also \(\gamma _{0} \in (0,1)\) such that \(\bar{c}_{\kappa} (1-\gamma ) \leq \frac{1}{2}\) for all \(\gamma \in (\gamma _{0},1)\). Therefore (76) and (77) are enough to show that
$$\begin{aligned} \bigl(v_{1}^{1}, v_{2}^{1} \bigr) \mapsto \bigl( \mathscr{N}_{1} \bigl(v_{1}^{1}, v_{2}^{1} \bigr), \mathscr{M}_{1} \bigl(v_{1}^{1},v_{2}^{1} \bigr) \bigr) \end{aligned}$$
is a contraction from the ball
$$\begin{aligned} \bigl\{ \bigl(v_{1}^{1}, v_{2}^{1} \bigr) \in {\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr) \times{\mathcal {C}}^{4, \alpha}_{\delta } \bigl( \mathbb{R}^{4} \bigr): \bigl\Vert \bigl(v_{1}^{1}, v_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2} \bigr\} \end{aligned}$$
into itself. Then applying a contraction mapping argument, we obtain the following proposition.
Proposition 7
Given \(\kappa >0\), \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma +\xi -1}{\xi}})\} )\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\) and \(\gamma _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\), for all \(\tau _{1}\) in some fixed compact subset of \([\tau _{1}^{-},\tau _{1}^{+}] \subset (0,\infty ) \) and for \(\varphi ^{1}_{j} \) and \(\psi ^{1}_{j}\) satisfying (65) and (72), there exists a unique \((v_{1}^{1}, v_{2}^{1}) (:= (v_{1,\varepsilon, \tau _{1}, \varphi ^{1}_{1}, \psi ^{1}_{1}}, v_{2,\varepsilon, \tau _{1}, \varphi ^{1}_{2}, \psi ^{1}_{2}}))\) solution of (71) such that
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{1}, v_{2}^{1} \bigr) \bigr\Vert _{C^{4, \alpha}_{\mu}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\delta}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x) := \frac{1}{\gamma}\bar{u}(x-x^{1}) - \frac{1-\gamma}{\gamma} G(\frac{\varepsilon x}{\tau _{1}},x^{2}) - \frac{1-\gamma }{\gamma \xi} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})\\ \phantom{v_{1}(x) := }{}- \frac{\ln \gamma}{\gamma} + h_{1}^{1}(x)+ H^{\mathrm{int}}(\varphi ^{1}_{1}, \psi ^{1}_{1}; {\frac{x-x^{1}}{R^{1}_{\varepsilon, \lambda}}}) + v_{1}^{1}(x), \\ v_{2}(x):= \frac{1}{\xi} G(\frac{\varepsilon x}{\tau _{1}}, x^{3})+ G(\frac{\varepsilon x}{\tau _{1}}, x^{2}) + h_{2}^{1}(x)+ H^{\mathrm{int}}( \varphi ^{1}_{2},\psi ^{1}_{2}; {\frac{x-x^{1}}{R^{1}_{\varepsilon, \lambda}}})+ v_{2}^{1}(x) \end{cases}\displaystyle \end{aligned}$$
solves (67) in \({B_{R_{\varepsilon, \lambda}^{1}}}(x^{1}) \).
In \(B_{R_{\varepsilon }^{3}}(x^{3})\), following the same arguments as the first case by reversing the roles of the functions \(v_{1}\) and \(v_{2}\) and by respecting the changes of the coefficients we can prove that there exists \((v_{1}^{3},v_{2}^{3})\in {\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\) such that
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{3}, v_{2}^{3} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
Furthermore, \((v_{1}^{3},v_{2}^{3})\) solves the equations
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} v_{1}^{3} = \frac{ 24 C_{3,\varepsilon }^{4 \frac{\gamma +\xi -1}{\xi }} 16^{\frac{1 - \gamma}{\xi} } \varepsilon ^{8 \frac {\gamma +\xi -1}{\xi}}}{\xi ^{\frac{1 - \gamma}{\xi}} (1+r^{2})^{ 4\frac{1 - \gamma}{\xi}}} \\ \phantom{\Delta ^{2} v^{1}_{2} =}{}\times e^{ \frac{ \gamma + \xi - 1}{\xi} G(\frac{\varepsilon x}{\tau _{3}}, x^{2}) + \frac{ \gamma + \xi - 1}{\gamma \xi} G( \frac{\varepsilon x}{\tau _{3}}, x^{1})+\gamma (h_{1}^{3} + H^{\mathrm{int}}_{ \varphi _{1}^{3}, \psi _{1}^{3} }+ v_{1}^{3} ) + (1 - \gamma ) (h_{2}^{3} + H^{\mathrm{int}}_{\varphi _{2}^{3}, \psi _{2}^{3} }+ v_{2}^{3} )} \\ \phantom{\Delta ^{2} v^{1}_{2} =}{}- \mathcal{L}_{\lambda} ({\frac{1}{\gamma}}G( \frac{\varepsilon x}{\tau _{3}}, x^{1}) +G( \frac{\varepsilon x}{\tau _{3}}, x^{2})\\ \phantom{\Delta ^{2} v^{1}_{2} =}{}+ H^{\mathrm{int}}(\varphi ^{3}_{1}, \psi ^{3}_{1};{\frac{x-x^{3}}{R_{\varepsilon,\lambda}^{3}}}) + h^{3}_{1}(x)+ v^{3}_{1}(x) ) -\Delta ^{2} h_{1}^{3}, \\ {\mathbb{L}} v_{2}^{3} = \frac{384 }{\xi (1 + r^{2})^{4}} [ e^{\xi (h_{2}^{3} + H^{\mathrm{int}}_{ \varphi _{2}^{3}, \psi _{2}^{3} } + v_{2}^{3} ) + (1 - \xi ) ( h_{1}^{3} + H^{\mathrm{int}}_{\varphi _{1}^{3}, \psi _{1}^{3} }+ v_{1}^{3} ) } - \xi v_{2}^{3} - 1 ] - \mathcal{L}_{\lambda} ({\frac{1}{\xi}}\bar{u}(x-x^{3}) \\ \phantom{{\mathbb{L}} v_{2}^{3} =}{}- {\frac{1-\xi}{\xi}} G(\frac{\varepsilon x}{\tau _{3}}, x^{2})- {\frac{1-\xi}{\gamma \xi}} G(\frac{\varepsilon x}{\tau _{3}}, x^{1})- \frac{\ln \xi}{\xi}\\ \phantom{{\mathbb{L}} v_{2}^{3} =}{}+ H^{\mathrm{int}}(\varphi ^{3}_{2}, \psi ^{3}_{2}; {\frac{x-x^{3}}{R_{\varepsilon, \lambda}^{3}}})+ h^{3}_{2}(x)+ v^{3}_{2}(x) ) - \Delta ^{2} h_{2}^{3}. \end{cases}\displaystyle \end{aligned}$$
(78)
Given \(\kappa >0\) (whose value will be fixed later), we further assume that the functions \(\varphi _{j}^{3}\) and \(\psi _{j}^{3}\) satisfy
$$\begin{aligned} \bigl\Vert \varphi _{j}^{3} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} \leq \kappa r_{ \varepsilon, \lambda}^{2} \quad\text{and} \quad\bigl\Vert \psi _{j}^{3} \bigr\Vert _{{ \mathcal {C}}^{2, \alpha}(S^{3})} \leq \kappa r_{\varepsilon, \lambda}^{2}, \quad\text{for } j=1, 2. \end{aligned}$$
(79)
Then we have the following proposition.
Proposition 8
Given \(\kappa >0\), \(\mu \in (1, 2)\) and \(\delta \in (0, \min \{({\frac{\gamma +\xi -1}{\gamma}}), ({\frac{\gamma +\xi -1}{\xi}})\} )\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\), and \(\xi _{0} \in (0,1)\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\xi \in (\xi _{0}, 1)\), for all \(\tau _{3}\) in some fixed compact subset of \([\tau _{3}^{-},\tau _{3}^{+}] \subset (0,\infty ) \) and for \(\varphi ^{3}_{j} \) and \(\psi ^{3}_{j}\) satisfying (65) and (79), there exists a unique \((v_{1}^{3}, v_{2}^{3}) (:= (v_{1,\varepsilon, \tau _{3}, \varphi ^{3}_{1}, \psi ^{3}_{1}}, v_{2,\varepsilon, \tau _{3}, \varphi ^{3}_{2}, \psi ^{3}_{2}}))\) solution of (71) such that
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{3}, v_{2}^{3} \bigr) \bigr\Vert _{C^{4, \alpha}_{\delta}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\mu}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x):= \frac{1}{\gamma} G(\frac{\varepsilon x}{\tau _{3}}, x^{1})+ G(\frac{\varepsilon x}{\tau _{3}}, x^{2}) + h_{1}^{3}(x)+ H^{\mathrm{int}}( \varphi ^{3}_{1},\psi ^{3}_{1},{\frac{x-x^{3}}{R^{3}_{\varepsilon, \lambda}}})+ v_{1}^{3}(x), \\ v_{2}(x) := \frac{1}{\xi}\bar{u}(x-x^{3}) -\frac{1-\xi}{\xi} G( \frac{\varepsilon x}{\tau _{3}}, x^{2}) -\frac{1-\xi}{\gamma \xi} G( \frac{\varepsilon x}{\tau _{3}}, x^{1})\\ \phantom{v_{2}(x) :=}{}-\frac{\ln \xi}{\xi} + h_{2}^{3}(x)+ H^{\mathrm{int}}(\varphi ^{3}_{2},\psi ^{3}_{2},{\frac{x-x^{3}}{R^{3}_{ \varepsilon, \lambda}}})+ v_{2}^{3}(x) \end{cases}\displaystyle \end{aligned}$$
solves (69) in \({B_{R_{\varepsilon, \lambda}^{3}}}(x^{3}) \).
In \(B_{R_{\varepsilon, \lambda}^{2}}(x^{2})\), we look for a solution of (45) of the form
$$\begin{aligned} \textstyle\begin{cases} v_{1} (x) = {\bar{u} }(x-x^{2}) + H^{\mathrm{int}}(\varphi _{1}^{2} , \psi _{1}^{2}; \frac{x - x^{2} }{R_{\varepsilon, \lambda}^{2}}) + h_{1}^{2}(x)+ v_{1}^{2}(x), \\ v_{2} (x) = {\bar{u} }(x-x^{2}) + H^{\mathrm{int}}(\varphi _{2}^{2} , \psi _{2}^{2}; \frac{x- x^{2} }{R_{\varepsilon, \lambda}^{2}}) + h_{2}^{2}(x)+ v_{2}^{2}(x). \end{cases}\displaystyle \end{aligned}$$
This amounts to solve the equations
$$\begin{aligned} \textstyle\begin{cases} {\mathbb{L}} v^{2}_{1} = \frac{384 }{(1 + r^{2})^{4}} [ e^{ \gamma (h^{2}_{1} + H^{\mathrm{int}}_{\varphi _{1}^{2}, \psi _{1}^{2} }+ v^{2}_{1} ) + (1 - \gamma ) (h^{2}_{2} + H^{\mathrm{int}}_{ \varphi _{2}^{2}, \psi _{2}^{2} }+ v^{2}_{2} )}-v^{2}_{1} - 1 ] \\ \phantom{{\mathbb{L}} v^{2}_{1} =}{}- \mathcal{L}_{\lambda} ({\bar{u} }(x-x^{2}) + H^{\mathrm{int}}_{ \varphi _{1}^{2}, \psi _{1}^{2}}( \frac{x - x^{2} }{R_{\varepsilon, \lambda}^{2}}) + h_{1}^{2}(x)+ v_{1}^{2}(x) ) - \Delta ^{2} h^{2}_{1}, \\ {\mathbb{L}} v^{2}_{2} =\frac{384 }{(1 + r^{2})^{4}} [e^{ \xi (h^{2}_{2} + H^{\mathrm{int}}_{\varphi _{2}^{2}, \psi _{2}^{2} }+ v^{2}_{2} ) + (1 - \xi ) (h^{2}_{1} + H^{\mathrm{int}}_{\varphi _{1}^{2} , \psi _{1}^{2}} + v^{2}_{1}} ) - v^{2}_{2} - 1 ] \\ \phantom{{\mathbb{L}} v^{2}_{2} =}{} - \mathcal{L}_{\lambda} ( {\bar{u} }(x-x^{2}) + H^{\mathrm{int}}_{\varphi _{2}^{2} , \psi _{2}^{2}} (\frac{x- x^{2} }{R_{\varepsilon, \lambda}^{2}}) + h_{2}^{2}(x)+ v_{2}^{2}(x) ) - \Delta ^{2} h^{2}_{2}. \end{cases}\displaystyle \end{aligned}$$
(80)
We denote by
$$\begin{aligned} {\mathbb{L}} v^{2}_{1} = \mathscr{R}_{3} \bigl(v_{1}^{2},v_{2}^{2} \bigr)\quad \text{and}\quad {\mathbb{L}} v^{2}_{2} = \mathscr{R}_{4} \bigl(v_{1}^{2},v_{2}^{2} \bigr). \end{aligned}$$
To find a solution of (80), it is enough to find a fixed point \((v_{1}^{2},v_{2}^{2})\) in a small ball of \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4})\times \mathcal{C}^{4, \alpha}_{\mu}(\mathbb{R}^{4})\), solutions of
$$\begin{aligned} \textstyle\begin{cases} v_{1}^{2} = \mathcal {G}_{\mu }\circ{\xi}_{{\mu}, R_{\varepsilon, \lambda}^{2}} \circ \mathscr{R}_{3}(h_{1}^{2},h_{2}^{2}) = \mathscr{N}_{2} (v_{1}^{2},v_{2}^{2}), \\ v_{2}^{2} = \mathcal {G}_{\mu }\circ{\xi}_{{\mu}, R_{\varepsilon, \lambda}^{2}} \circ \mathscr{R}_{4}(v_{1}^{2},v_{2}^{2}) = \mathscr{M}_{2} (v_{1}^{2},v_{2}^{2}). \end{cases}\displaystyle \end{aligned}$$
(81)
Given \(\kappa >0\) (whose value will be fixed later), we further assume that the functions \(\varphi _{j}^{2}\) and \(\psi _{j}^{2}\) satisfy
$$\begin{aligned} \bigl\Vert \varphi _{j}^{2} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} \leq \kappa r_{ \varepsilon, \lambda}^{2} \quad\text{and}\quad \bigl\Vert \psi _{j}^{2} \bigr\Vert _{{ \mathcal {C}}^{2, \alpha}(S^{3})} \leq \kappa r_{\varepsilon, \lambda}^{2}, \quad\text{for } j=1, 2. \end{aligned}$$
(82)
Then, we have the following result.
Lemma 6
Let \(\mu \in (1,2)\), \(\gamma _{0} \textit{ and } \xi _{0}\in (0,1)\). Given \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\), \(c_{\kappa} > 0\) and \(\bar{c}_{\kappa} > 0\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0},1)\) and \(\xi \in (\xi _{0},1)\). We have
$$\begin{aligned} &\bigl\Vert \mathcalligra{ }\mathscr{N}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})}\leq c_{\kappa } r_{\varepsilon, \lambda}^{2},\qquad \bigl\Vert \mathcalligra{ }\mathscr{M}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4} )}\leq c_{\kappa } r_{\varepsilon, \lambda}^{2},\\ &\bigl\Vert \mathcalligra{ }\mathscr{N}_{2} \bigl(v_{1}^{2},v_{2}^{2} \bigr) -\mathcalligra{ }\mathscr{N}\mathcalligra{ }_{2} \bigl(t_{1}^{2},t_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\\ &\quad\leq \bar{c}_{ \kappa } \bigl(1 - \gamma +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{2}-t_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}\\ &\bigl\Vert \mathscr{M}_{2} \bigl(v_{1}^{2},v_{2}^{2} \bigr) -\mathcalligra{ }\mathscr{M}\mathcalligra{ }_{2} \bigl(t_{1}^{2},t_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})}\\ &\quad \leq \bar{c}_{ \kappa }(1 - \xi ) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa} \bigl(1 - \xi +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert v_{2}^{2}-t_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}, \end{aligned}$$
provided \((v_{1}^{2}, v_{2}^{2})\), \((t_{1}^{2}, t_{2}^{2})\) in \(\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^{4}) \times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\) satisfying
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{2}, v_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa }r_{\varepsilon, \lambda}^{2} \quad\textit{and}\quad \bigl\Vert \bigl(t_{1}^{2}, t_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})\times{ \mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa }r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
(83)
Proof
The proof of the first and the second estimates follows from the asymptotic behavior of \(H^{\mathrm{int}}\) together with the assumption on the norms of \(\varphi ^{2}_{j}\) and \(\psi ^{2}_{j}\) given by (82) and it follows from the estimate of \(H^{\mathrm{int}}\), given by Lemma 3, that
$$\begin{aligned} \biggl\Vert H^{\mathrm{int}}_{\varphi ^{2}_{j}, \psi ^{2}_{j}} \biggl({\frac{r}{R^{2}_{ \varepsilon, \lambda}}} \cdot \biggr) \biggr\Vert _{{\mathcal {C}}^{4, \alpha} ( \bar{B}_{2}(0) - B_{1}(0))} \leq C r^{2} \bigl(R^{2}_{\varepsilon, \lambda} \bigr)^{-2} \bigl( \bigl\Vert \varphi ^{2}_{j} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} + \bigl\Vert \psi ^{2}_{j} \bigr\Vert _{{\mathcal {C}}^{2, \alpha}(S^{3})} \bigr), \end{aligned}$$
for all \(r \leq {\frac{R^{2}_{\varepsilon, \lambda}}{2}}\). Then by (82), we get
$$\begin{aligned} \biggl\Vert H^{\mathrm{int}}_{\varphi ^{2}_{j}, \psi ^{2}_{j}} \biggl({\frac{r}{R^{2}_{ \varepsilon, \lambda}}} \cdot \biggr) \biggr\Vert _{{\mathcal {C}}^{4, \alpha} ( \bar{B}_{2}(0) - B_{1}(0))} \leq c_{\kappa }\varepsilon ^{2} r^{2}. \end{aligned}$$
On the other hand,
$$\begin{aligned} & \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4- \mu} \bigl\vert \mathscr{R}_{3}(0,0) \bigr\vert \\ &\quad \leq \sup _{r \leq R_{ \varepsilon, \lambda}^{2}} \frac{384 r^{4-\mu} }{(1 + r^{2})^{4}} \bigl\vert e^{ \gamma (h^{2}_{1} + H^{\mathrm{int}}_{\varphi _{1}^{2}, \psi _{1}^{2} } ) + (1 - \gamma ) (h^{2}_{2} + H^{\mathrm{int}}_{\varphi _{2}^{2}, \psi _{2}^{2} } )} - 1 \bigr\vert \\ & \qquad{}+ \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4- \mu} \bigl( \bigl\vert \mathcal{L}_{\lambda} \bigl(\bar{u} + H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} + h^{2}_{1} \bigr) \bigr\vert + \bigl\vert \Delta ^{2}h^{1}_{2} \bigr\vert \bigr) \\ & \quad\leq c_{\kappa }\sup_{r \leq R_{\varepsilon, \lambda}^{2}} \frac{ 384 r^{4-\mu} }{(1+r^{2})^{4}} \bigl( \gamma r^{2} \bigl\Vert H^{\mathrm{int}}_{\varphi _{1}^{2}, \psi _{1}^{2} } \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{2}} + \gamma \bigl\Vert h^{2}_{1} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu}} \\ &\qquad{}+ (1 - \gamma ) r^{2} \bigl\Vert H^{\mathrm{int}}_{\varphi _{2}^{2}, \psi _{2}^{2} } \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{2}}+ (1 - \gamma ) \bigl\Vert h^{2}_{2} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu}} \bigr) \\ &\qquad{}+\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl[ \bigl\vert \bigl(\Delta \bar{u}+ \Delta H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} + \Delta h^{2}_{1} \bigr)^{2} \bigr\vert \\ &\qquad{}+ 2 \bigl\vert \nabla \bigl(\bar{u} + H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} + h^{2}_{1} \bigr) \nabla \bigl(\Delta \bigl(\bar{u}+ H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} + h^{2}_{1} \bigr) \bigr) \bigr\vert \bigr] \\ &\qquad{} + \lambda ^{2} \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \nabla \bigl(\bar{u} + H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} + h^{2}_{1} \bigr) \bigr\vert ^{2} \bigl\vert \Delta \bigl( \bar{u}+ H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} + h^{2}_{1} \bigr) \bigr\vert +\sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \Delta ^{2} h^{2}_{1} \bigr\vert \\ &\quad \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{\kappa } \lambda \bigl(1 + \varepsilon ^{\mu }r_{\varepsilon, \lambda}^{2-\mu} + r_{ \varepsilon, \lambda}^{2} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{\mu +4} \bigr) + c_{\kappa } \lambda ^{2} \bigl(1 + \varepsilon ^{ \mu }r_{\varepsilon, \lambda}^{2-\mu} + \varepsilon ^{\mu +2} r_{ \varepsilon, \lambda}^{6-\mu} \\ &\qquad{} + \varepsilon ^{\mu +1} r_{\varepsilon, \lambda}^{4- \mu} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{\mu +4} + r_{ \varepsilon, \lambda}^{4} + \varepsilon ^{-2\mu} r_{\varepsilon, \lambda}^{2\mu +6} \bigr). \end{aligned}$$
Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get
$$\begin{aligned} \bigl\Vert \mathscr{N}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(84)
For the second estimate, we use the same techniques to prove
$$\begin{aligned} \bigl\Vert \mathscr{M}_{2}(0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(85)
To derive the third estimate, using the asymptotic behavior of \(H^{\mathrm{int}}\) given by the estimate (74) and for \(l=1,\ldots,4+\alpha \), \(\vert \nabla ^{l}w \vert \leq c_{\kappa }r^{\mu - l} \Vert w\Vert _{{ \mathcal {C}}_{\mu}^{k,\alpha}(\mathbb{R}^{4})}\). Then for \((v_{1}^{2},v_{2}^{2}), (t_{1}^{2},t_{2}^{2})\) verifying (58), we have
$$\begin{aligned} & \sup_{r\leq R_{\varepsilon, \lambda}^{2}} r^{4 - \mu} \bigl\vert \mathscr{R}_{3} \bigl(v_{1}^{2},v_{2}^{2} \bigr) -\mathscr{R}_{3} \bigl(t_{1}^{2},t_{2}^{2} \bigr) \bigr\vert \\ &\quad\leq \sup_{r\leq R_{\varepsilon, \lambda}^{2}} \frac{384r^{4-\mu}}{(1+r^{2})^{4}} \bigl\vert \bigl( e^{\gamma ( h_{1}^{2} + H^{\mathrm{int}}_{\varphi _{1}^{2}, \psi _{1}^{2} }+ v_{1}^{2} ) + (1 - \gamma ) ( h_{2}^{2} + H^{\mathrm{int}}_{\varphi _{2}^{2}, \psi _{2}^{2} }+ v_{2}^{2} ) } - v_{1}^{2} \bigr) \\ &\qquad{}- \bigl( e^{\gamma (h_{1}^{2} + H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1} } + t_{1}^{2} ) + (1 - \gamma ) (h_{2}^{2} + H^{\mathrm{int}}_{\varphi _{2}^{2}, \psi _{2}^{2} } + t_{2}^{2} ) } - t_{1}^{2} \bigr) \bigr\vert \\ &\qquad{}+ \sup _{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \mathcal{L}_{\lambda} \bigl(\bar{u} + H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1} +v^{2}_{1} \bigr) - \mathcal{L}_{\lambda} \bigl(\bar{u} + H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1} +t^{2}_{1} \bigr) \bigr\vert \\ &\quad \leq c \sup_{r\leq R_{\varepsilon, \lambda}^{2}} \frac{384 r^{4-\mu}}{(1+r^{2})^{4}} \bigl\vert (\gamma -1) \bigl(v_{1}^{2} -t_{1}^{2} \bigr) + (1 - \gamma ) \bigl(v_{2}^{2} - t_{2}^{2} \bigr) \bigr\vert +\lambda \sup_{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \Delta \bigl(v^{2}_{1}-t^{2}_{1} \bigr) \bigr\vert \\ &\qquad{}\times \bigl(2 \vert \Delta {\bar{u}} \vert +2 \bigl\vert \Delta H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} \bigr\vert + 2 \bigl\vert \Delta h^{2}_{1} \bigr\vert + \bigl\vert \Delta v^{2}_{1} \bigr\vert + \bigl\vert \Delta t^{2}_{1} \bigr\vert \bigr)\\ &\qquad{} +2 \lambda \sup _{r \leq R_{\varepsilon, \lambda}^{1}} r^{4-\mu} \bigl\vert \nabla \bigl(v^{2}_{1}-t^{2}_{1} \bigr) \cdot \nabla \bigl(2\Delta {\bar{u}} +2 \Delta H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} + 2 \Delta h^{2}_{1} \\ &\qquad{} + \Delta v^{2}_{1} + \Delta t^{2}_{1} \bigr) +\nabla \bigl(\Delta \bigl(v^{2}_{1}-t^{2}_{1} \bigr) \bigr) \cdot \nabla \bigl(2 {\bar{u}} +2 H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} + 2h^{2}_{1} + v^{2}_{1} + t^{2}_{1} \bigr) \bigr\vert \\ &\qquad{}+\lambda ^{2} \sup _{r \leq R_{\varepsilon, \lambda}^{2}} r^{4-\mu} \bigl\vert \Delta \bigl(v^{2}_{1}-t^{2}_{1} \bigr) \bigr\vert \\ &\qquad{}\times \bigl[ \bigl\vert \nabla \bigl(\bar{u} \bigl(x-x^{2} \bigr) + H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1}+v^{2}_{1} \bigr) \bigr\vert ^{2} + \bigl\vert \nabla \bigl(\bar{u} +H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1}+t^{2}_{1} \bigr) \bigr\vert ^{2} \bigr] \\ &\qquad{}+ \lambda ^{2} \sup _{r \leq R_{ \varepsilon, \lambda}^{2}} r^{4-\mu} \biggl(\frac{2}{\gamma} \vert \Delta {\bar{u}} \vert \\ &\qquad{} +2 \bigl\vert \Delta H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} \bigr\vert + 2 \bigl\vert \Delta h^{2}_{1} \bigr\vert + \bigl\vert \Delta v^{2}_{1} \bigr\vert + \bigl\vert \Delta t^{2}_{1} \bigr\vert \biggr) \bigl[ \bigl\vert \nabla \bigl(\bar{u} + H^{\mathrm{int}}_{ \varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1}+ v^{2}_{1} \bigr) \bigr\vert ^{2} \\ &\qquad{}- \bigl\vert \nabla \bigl(\bar{u} + H^{\mathrm{int}}_{\varphi ^{2}_{1}, \psi ^{2}_{1}} +h^{2}_{1} +t^{2}_{1} \bigr) \bigr\vert ^{2} \bigr] \\ &\quad\leq c_{\kappa} (1 - \gamma ) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{ \mu}^{4,\alpha}} + {c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{2}-t_{2}^{2} \bigr\Vert _{{ \mathcal {C}}_{\mu}^{4,\alpha}}\\ &\qquad{} + {c}_{\kappa } \lambda \bigl( 1 + r_{\varepsilon, \lambda}^{2} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} \bigr) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}} \\ &\qquad{} + {c}_{\kappa} \lambda ^{2} \bigl( 1 + r_{ \varepsilon, \lambda}^{2} + \varepsilon ^{-\mu} r_{\varepsilon, \lambda}^{2+\mu} + \varepsilon ^{-2\mu} r_{\varepsilon, \lambda}^{2(2+ \mu )} \bigr) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}}. \end{aligned}$$
Making use of Proposition 1 together with (38) and using the condition \((A_{1})\) for \(\mu \in (1,2)\), we get that there exists \(\bar{c}_{\kappa} > 0\) such that
$$\begin{aligned} &\bigl\Vert \mathcalligra{ }\mathscr{N}_{2} \bigl(v_{1}^{2},v_{2}^{2} \bigr) -\mathcalligra{ }\mathscr{N}\mathcalligra{ }_{2} \bigl(t_{1}^{2},t_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad\leq \bar{c}_{ \kappa } \bigl(1 - \gamma +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa}(1 - \gamma ) \bigl\Vert v_{2}^{2}-t_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(86)
Similarly, we get
$$\begin{aligned} &\bigl\Vert \mathscr{M}_{2} \bigl(v_{1}^{2},v_{2}^{2} \bigr) -\mathscr{M}_{2} \bigl(t_{1}^{2},t_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \\ &\quad \leq \bar{c}_{ \kappa }(1 - \xi ) \bigl\Vert v^{2}_{1}-t^{2}_{1} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4, \alpha}(\mathbb{R}^{4})} + \bar{c}_{\kappa} \bigl(1 - \xi +r_{\varepsilon, \lambda}^{2} \bigr) \bigl\Vert v_{2}^{2}-t_{2}^{2} \bigr\Vert _{{\mathcal {C}}_{\mu}^{4,\alpha}( \mathbb{R}^{4})}. \end{aligned}$$
(87)
□
Then there exist \(\gamma _{0} \text{ and } \xi _{0} \in (0,1)\), reducing \(\varepsilon _{\kappa} \) and \(\lambda _{\kappa} \), if necessary, we can assume that \(\bar{c}_{\kappa } (1 - \gamma +r_{\varepsilon, \lambda}^{2})\leq 1/2\) and \(\bar{c}_{\kappa } (1 - \xi +r_{\varepsilon, \lambda}^{2})\leq 1/2\) for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\) and \(\xi \in (\xi _{0}, 1)\). Therefore (61) and (62) are enough to show that
$$\begin{aligned} \bigl(v_{1}^{2}, v_{2}^{2} \bigr) \mapsto \bigl( \mathscr{N}_{2} \bigl(v_{1}^{2}, v_{2}^{2} \bigr), \mathscr{M}_{2} \bigl(v_{1}^{2},v_{2}^{2} \bigr) \bigr) \end{aligned}$$
is a contraction from the ball
$$\begin{aligned} \bigl\{ \bigl(v_{1}^{2}, v_{2}^{2} \bigr) \in {\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr) \times{\mathcal {C}}^{4, \alpha}_{\mu } \bigl( \mathbb{R}^{4} \bigr): \bigl\Vert \bigl(v_{1}^{2}, v_{2}^{2} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2} \bigr\} \end{aligned}$$
into itself. Then applying a contraction mapping argument, we obtain the following proposition.
Proposition 9
Given \(\kappa >0\), \(\mu \in (1, 2)\), \(\gamma _{0} \in (0,1)\) and \(\xi _{0} \in (0,1)\), there exist \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\) and \(c_{\kappa} > 0\) such that for all \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\gamma \in (\gamma _{0}, 1)\) and \(\xi \in (\xi _{0}, 1)\), for all \(\tau _{2}\) in some fixed compact subset of \([\tau _{2}^{-},\tau _{2}^{+}] \subset (0,\infty ) \) and for \(\varphi ^{2}_{j} \) and \(\psi ^{2}_{j}\) satisfying (65) and (82), there exists a unique \((v_{1}^{2}, v_{2}^{2}) (:= (v_{1,\varepsilon, \tau _{2}, \varphi ^{2}_{1}, \psi ^{2}_{1}}, v_{2,\varepsilon, \tau _{2}, \varphi ^{2}_{2}, \psi ^{2}_{2}}))\) solution of (81) such that
$$\begin{aligned} \bigl\Vert \bigl(v_{1}^{2}, v_{2}^{2} \bigr) \bigr\Vert _{C^{4, \alpha}_{\mu}(\mathbb{R}^{4}) \times C^{4, \alpha}_{\mu}(\mathbb{R}^{4})} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
Hence
$$\begin{aligned} \textstyle\begin{cases} v_{1}(x) := \bar{u}(x-x^{2}) + h_{1}^{2}(x)+ H^{\mathrm{int}}( \varphi ^{2}_{1},\psi ^{2}_{1}; {\frac{x-x^{2}}{R^{2}_{\varepsilon, \lambda}}})+ v_{1}^{2}(x), \\ v_{2}(x):= \bar{u}(x-x^{2}) + h_{2}^{2}(x)+ H^{\mathrm{int}}( \varphi ^{2}_{2},\psi ^{2}_{2}; {\frac{x-x^{2}}{R^{2}_{\varepsilon, \lambda}}})+ v_{2}^{2}(x), \end{cases}\displaystyle \end{aligned}$$
solves (68) in \({B_{R_{\varepsilon, \lambda}^{2}}}(x^{2}) \).
Remark also that the functions \((v_{1}^{i}, v_{2}^{i})(:= (v^{i}_{1,\varepsilon, \tau _{i}, \varphi ^{i}_{1}, \psi ^{i}_{1}}, v^{i}_{2,\varepsilon, \tau _{i}, \varphi ^{i}_{2}, \psi ^{i}_{2}})) \), for \(i\in \{1,2,3\}\), depend continuously on the parameter \(\tau _{i}\).
3.3 The nonlinear exterior problem
Given \({\tilde{\mathbf {x}}}:= (\tilde {x}^{1}, \tilde {x}^{2}, \tilde {x}^{3}) \in \Omega ^{3}\) close to \({\mathbf{x}}: = (x^{1}, x^{2},x^{3})\), \({\boldsymbol{\eta}}:= (\eta _{1}, \eta _{2}, \eta _{3})\in {\mathbb{R}}\) close to 0, \({\tilde{\mathbf {\varphi}}}_{1}: = (\tilde{\varphi}^{1}_{1}, \tilde{\varphi}^{2}_{1},\tilde{\varphi}^{3}_{1} ) \in ({\mathcal {C}}^{4, \alpha} (S^{3}))^{3}\), \({\tilde{\mathbf {\varphi}}}_{2}: = (\tilde{\varphi}^{1}_{2}, \tilde{\varphi}^{2}_{2},\tilde{\varphi}^{3}_{2} ) \in ({\mathcal {C}}^{4, \alpha} (S^{3}))^{3}\), \({\tilde{\mathbf {\psi}}}_{1}: = (\tilde{\psi}^{1}_{1}, \tilde{\psi}^{2}_{1}, \tilde{\psi}^{3}_{1} ) \in ({\mathcal {C}}^{2, \alpha} (S^{3}))^{3}\) and \({\tilde{\mathbf {\psi}}}_{2}: = (\tilde{\psi}^{1}_{2}, \tilde{\psi}^{2}_{2}, \tilde{\psi}_{2}^{3} ) \in ({\mathcal {C}}^{2, \alpha} (S^{3}))^{3}\) satisfying (66). Let \({\tilde{\mathbf {w}_{1}}}\) and \({\tilde{\mathbf {w}_{2}}}\) be defined by
$$\begin{aligned} \begin{aligned} {\tilde{\mathbf {w}_{1}}}(x):={}& \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &{}+ \sum _{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}} \biggl( \tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}; {\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \quad \text{and} \\ {\tilde{\mathbf {w}_{2}}}(x):={}& \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &{}+ \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}} \biggl( \tilde {\varphi}_{2}^{i}, \tilde { \psi}_{2}^{i}; {\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr). \end{aligned} \end{aligned}$$
(88)
Here \(\chi _{r_{0}}\) is a cut-off function identically equal to 1 in \(B_{{\frac{r_{0}}{2}}}(0)\) and identically equal to 0 outside \(B_{r_{0}}(0)\). We would like to find a solution of the system
$$\begin{aligned} \Delta ^{2} u_{1} = \rho ^{4} e^{\gamma u_{1} + (1-\gamma ) u_{2}} - \mathcal{L}_{\lambda}(u_{1}) \quad\text{and}\quad \Delta ^{2} u_{2} = \rho ^{4} e^{\xi u_{2} +(1-\xi ) u_{1}} -\mathcal{L}_{\lambda}(u_{2}) \end{aligned}$$
(89)
in the domain \(\bar{\Omega}_{r_{\varepsilon, \lambda}}(\tilde{\mathbf {x})} \), which is a perturbation of \({\tilde{\mathbf {w}_{k}}}\), \(k=1,2\), with
$$\begin{aligned} \mathcal{L}_{\lambda}(u_{i})=\lambda (\Delta u_{i})^{2} + \lambda \nabla u_{i} \cdot \nabla ( \Delta u_{i})+ \lambda ^{2} \vert \nabla u_{i} \vert ^{2} \Delta u_{i}, \quad\text{for } i=1,2. \end{aligned}$$
Writing \(v_{k}= \tilde{\mathbf{w}}_{k} + \tilde{v}_{k}\), this amounts to solve in \(\bar{\Omega}_{r_{\varepsilon, \lambda}}(\tilde{\mathbf {x})}\)
$$\begin{aligned} \textstyle\begin{cases} \Delta ^{2} \tilde{v}_{1} = \rho ^{4} e^{\gamma ( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1}) + (1-\gamma ) ( \tilde{\mathbf{w}}_{2} + \tilde{v}_{2})} - \mathcal{L}_{\lambda}( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1})- \Delta ^{2} \tilde{\mathbf{w}}_{1}, \\ \Delta ^{2} \tilde{v}_{2} = \rho ^{4} e^{ \xi ( \tilde{\mathbf{w}}_{2} + \tilde{v}_{2}) +(1- \xi ) ( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1})} - \mathcal{L}_{\lambda}( \tilde{\mathbf{w}}_{2} + \tilde{v}_{2})- \Delta ^{2} \tilde{\mathbf{w}}_{2}. \end{cases}\displaystyle \end{aligned}$$
(90)
For all \(\sigma \in (0,{\frac{r_{0}}{2}}) \) and all \({\tilde{\mathbf {x}}} =(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3}) \in \Omega ^{3} \) such that \(\|{\mathbf{x}} - {\tilde{\mathbf {x}}} \| \leq {\frac{r_{0}}{2}}\), where \({\mathbf{x}} = (x^{1}, x^{2}, x^{3})\), we denote by \(\tilde{\xi}_{\sigma, {\tilde{\mathbf {x}}}}: {\mathcal {C}}^{0, \alpha}_{ \nu} (\bar{\Omega}_{\sigma} ({\tilde{\mathbf {x}}})) \longrightarrow { \mathcal {C}}^{0, \alpha}_{\nu }({\bar{\Omega}}^{*}({\tilde{\mathbf {x}}}))\) the extension operator defined by
$$\begin{aligned} \textstyle\begin{cases} \tilde{\xi}_{\sigma, {\tilde{\mathbf {x}}}}(f) \equiv f &\text{in } \bar{\Omega}_{\sigma}({\tilde{\mathbf {x}}}), \\ \tilde{\xi}_{\sigma, {\tilde{\mathbf {x}}}} (f)(\tilde{x}^{j}+x) = \tilde{\chi} (\frac{ \vert x \vert }{\sigma} ) f ( \tilde{x}^{j} + \sigma \frac{x}{ \vert x \vert } ) & \text{in } B_{\sigma}( \tilde{x}^{j}) - B_{\frac{\sigma}{2}}(\tilde{x}^{j}) \ \forall 1 \leq j\leq 3, \\ \tilde{\xi}_{\sigma, {\tilde{\mathbf {x}}}} (f) \equiv 0 &\text{in } B_{ \frac{\sigma}{2}}(\tilde{x}^{1})\cup B_{\frac{\sigma}{2}}(\tilde{x}^{2}) \cup B_{\frac{\sigma}{2}}(\tilde{x}^{3}). \end{cases}\displaystyle \end{aligned}$$
Here χ̃ is a cut-off function over \(\mathbb{R}_{+}\), which is equal to 1 for \(t \geq 1 \) and equal to 0 for \(t \leq {\frac{1}{2}} \). Obviously, there exists a constant \(\bar{c} =\bar{c}(\nu ) > 0 \) only depending on ν such that
$$\begin{aligned} \bigl\Vert \tilde{\xi}_{\sigma, {\tilde{\mathbf {x}}}}(w) \bigr\Vert _{{ \mathcal {C}}^{0, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))} \leq \bar{c} \Vert w \Vert _{{\mathcal {C}}^{0, \alpha}_{\nu}(\bar{\Omega}_{ \sigma}({\tilde{\mathbf {x}}}))}. \end{aligned}$$
(91)
We fix \(\nu \in (-1,0)\), to solve (90), it is enough to find \((\tilde{v}_{1}, \tilde{v}_{2}) \in ({\mathcal {C}}^{4, \alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2} \) solution of
$$\begin{aligned} \tilde{v}_{1} = \widetilde{\mathcal {K}}_{\nu} \circ {\tilde{\xi}}_{r_{ \varepsilon, \lambda}, {\tilde{\mathbf {x}}}} \circ \tilde{S}_{1}( \tilde{v}_{1}, \tilde{v}_{2}) \quad\text{and}\quad \tilde{v}_{2} = \widetilde{\mathcal {K}}_{\nu} \circ {\tilde{\xi}}_{r_{\varepsilon, \lambda}, {\tilde{\mathbf {x}}}} \circ \tilde{S}_{2}(\tilde{v}_{1}, \tilde{v}_{2}), \end{aligned}$$
(92)
where
$$\begin{aligned} \textstyle\begin{cases} \tilde{S}_{1} (\tilde{v}_{1}, \tilde{v}_{2}) := \rho ^{4} e^{ \gamma (\tilde{\mathbf{w}}_{1} + \tilde{v}_{1}) + (1-\gamma ) ( \tilde{\mathbf{w}}_{2} + \tilde{v}_{2})} - \mathcal{L}_{\lambda}( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1})- \Delta ^{2} \tilde{\mathbf{w}}_{1}, \\ \tilde{S}_{2} (\tilde{v}_{1}, \tilde{v}_{2}) := \rho ^{4} e^{ \xi (\tilde{\mathbf{w}}_{2} + \tilde{v}_{2}) +(1- \xi ) ( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1})} - \mathcal{L}_{\lambda}( \tilde{\mathbf{w}}_{2} + \tilde{v}_{2})- \Delta ^{2} \tilde{\mathbf{w}}_{2}. \end{cases}\displaystyle \end{aligned}$$
We denote by
$$\begin{aligned} \tilde {\mathcal{N}}(\tilde{v}_{1}, \tilde{v}_{2}):= \widetilde{{\mathcal {K}}}_{\nu} \circ {\tilde{\xi}}_{r_{ \varepsilon, \lambda}, {\tilde{\mathbf {x}}}} \circ \tilde{S}_{1}( \tilde{v}_{1}, \tilde{v}_{2}) \quad\text{and}\quad \tilde {\mathcal{M}}( \tilde{v}_{1}, \tilde{v}_{2}):= \widetilde{{\mathcal {K}}}_{\nu} \circ {\tilde{\xi}}_{r_{ \varepsilon, \lambda}, {\tilde{\mathbf {x}}}} \circ \tilde{S}_{2}( \tilde{v}_{1}, \tilde{v}_{2}). \end{aligned}$$
Given \(\kappa > 0\) (whose value will be fixed later), we further assume that for \(i\in \{1, 2, 3\}\) and \(j\in \{1, 2\}\) the functions \(\tilde{\varphi}^{i}_{j}\), \(\tilde{\psi}^{i}_{j}\), the parameters \(\eta _{i}\) and the point \({\tilde{\mathbf {x}}} = (\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})\) satisfy
$$\begin{aligned} & \bigl\Vert \tilde{\varphi}^{i}_{j} \bigr\Vert _{{\mathcal {C}}^{4, \alpha}(S^{3})} \leq \kappa r_{\varepsilon, \lambda}^{2} ,\qquad \bigl\Vert \tilde{\psi}^{i}_{j} \bigr\Vert _{{\mathcal {C}}^{2, \alpha}(S^{3})} \leq \kappa r_{\varepsilon, \lambda}^{2}, \end{aligned}$$
(93)
$$\begin{aligned} &\vert \eta _{i} \vert \leq \kappa r_{\varepsilon, \lambda}^{2} ,\qquad \bigl\vert \tilde{x}^{i} - x^{i} \bigr\vert \leq \kappa r_{\varepsilon, \lambda}. \end{aligned}$$
(94)
Then the following result holds.
Lemma 7
Under the above assumptions, there exists a constant \(c_{\kappa }>0\) such that
$$\begin{aligned} &\bigl\Vert \tilde{\mathcal{N}} (0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})) } \leq c_{\kappa } r_{ \varepsilon, \lambda}^{2},\\ &\bigl\Vert \tilde{\mathcal{M}} (0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})) } \leq c_{\kappa } r_{ \varepsilon, \lambda}^{2},\\ &\bigl\Vert \tilde {\mathcal{N}} (\tilde {v}_{1}, \tilde {v}_{2}) - \tilde {\mathcal{N }} \bigl( \tilde {v}_{1}', \tilde {v}_{2}' \bigr) \bigr\Vert _{{ \mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )} \leq {c}_{\kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert ( \tilde {v}_{1}, \tilde {v}_{2})- \bigl( \tilde {v}_{1}', \tilde {v}_{2}' \bigr) \bigr\Vert _{({\mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2}} \end{aligned}$$
and
$$\begin{aligned} \bigl\Vert \tilde {\mathcal{M}} (\tilde{v}_{1}, \tilde{v}_{2}) - \tilde {\mathcal{M}} \bigl( \tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )} \leq {c}_{ \kappa} r_{\varepsilon, \lambda}^{2} \bigl\Vert (\tilde{v}_{1}, \tilde{v}_{2})- \bigl( \tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\Vert _{({\mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2}}, \end{aligned}$$
provided \((\tilde{v}_{1}, \tilde{v}_{2},\tilde{v}_{1}', \tilde{v}_{2}' ) \in ({\mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) )^{4}\) satisfy
$$\begin{aligned} \| ({\tilde{v}}_{1},{\tilde{v}}_{2}) \|_{({\mathcal {C}}^{4, \alpha}_{ \nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2}} \leq 2 c_{ \kappa } r_{\varepsilon, \lambda}^{2}\quad \textit{and} \quad\bigl\Vert \bigl( \tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\Vert _{({\mathcal {C}}^{4, \alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2}} \leq 2 c_{\kappa } r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
(95)
Proof
As for the interior problem, the proof of the two first estimates follows from the asymptotic behavior of \(H^{\mathrm{ext}} \) together with the assumption on the norm of boundary data \(\tilde {\varphi}^{i}_{j} \) and \(\tilde {\psi}^{i}_{j} \), given by (93). Indeed, let \(c_{\kappa} \) be a constant depending only on κ, by Lemma 3,
$$\begin{aligned} \biggl\vert H^{\mathrm{ext}} \biggl(\tilde{\varphi}^{i}_{j}, \tilde{\psi}^{i}_{j}; \frac{x - \tilde{x}^{i}}{r_{\varepsilon, \lambda}} \biggr) \biggr\vert \leq c_{ \kappa } r_{\varepsilon, \lambda}^{3} r^{-1}. \end{aligned}$$
(96)
On the other hand,
$$\begin{aligned} &\tilde{S}_{1}(0,0) = \rho ^{4} e^{\gamma \tilde{\mathbf {w}_{1}}+\mathbf{(1}-\boldsymbol{ \gamma ) }\tilde{\mathbf {w}_{2}}}- \mathcal{L}_{\lambda}(\tilde{\mathbf{w}}_{1}) - \Delta ^{2} {\tilde{\mathbf {w}_{1}}} \quad\text{and}\\ & \tilde{S}_{2}(0,0) = \rho ^{4} e^{\xi \tilde{\mathbf {w}_{2}}+\mathbf{ (1}-\boldsymbol{\xi ) }\tilde{\mathbf {w}_{1}}} - \mathcal{L}_{\lambda}(\tilde{\mathbf{w}}_{2}) - \Delta ^{2} {\tilde{\mathbf {w}_{2}}}. \end{aligned}$$
We will estimate \(\tilde{S}_{1}(0,0)\) in different subregions of \(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) \).
• In \(B_{{\frac{r_{0}}{2}}}(\tilde{x}^{1})- B_{r_{\varepsilon, \lambda}}( \tilde{x}^{1})\), we have \(\chi _{r_{0}}(x - \tilde{x}^{1}) = 1, \chi _{r_{0}}(x - \tilde{x}^{2}) = 0, \chi _{r_{0}}(x - \tilde{x}^{3}) = 0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{1}}}= 0\), so that \(|\tilde{S}_{1}(0,0)| = \rho ^{4} e^{\gamma \tilde{w}_{1} + (1- \gamma ) \tilde{w}_{2}} - \mathcal{L}_{\lambda}(\tilde{\mathbf{w}}_{1})\). Then
$$\begin{aligned} & \bigl\vert \tilde{S}_{1}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa} \varepsilon ^{4} \bigl\vert x - \tilde{x}^{1} \bigr\vert ^{-8 (1 + \eta _{1})} \\ &\qquad{}+\lambda \biggl\vert \biggl( \Delta \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{1}, \tilde {\psi}_{1}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + \lambda \biggl\vert \nabla \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{1}, \tilde {\psi}_{1}^{1}} \biggl({ \frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \cdot \nabla \biggl(\Delta \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{1}, \tilde {\psi}_{1}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr) \biggr\vert \\ &\qquad{} + \lambda ^{2} \biggl\vert \nabla \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{1}, \tilde {\psi}_{1}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \Delta \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{1}, \tilde {\psi}_{1}^{1}} \biggl({ \frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert \\ &\quad\leq c_{\kappa} \varepsilon ^{4} r^{-8 (1 + \eta _{1})} + c_{ \kappa} \lambda \biggl\vert \biggl(\frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-2}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-2} + r_{\varepsilon, \lambda}^{3} r^{-3} \biggr)^{2} \biggr\vert \\ &\qquad{}+ c_{\kappa} \lambda \biggl\vert \biggl(\frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-1}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-1} + r_{\varepsilon, \lambda}^{3} r^{-2} \biggr)\\ &\qquad{}\times \biggl(\frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-3}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-3} + r_{\varepsilon, \lambda}^{3} r^{-4} \biggr) \biggr\vert \\ &\qquad{}+ c_{ \kappa} \lambda ^{2} \biggl\vert \frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-1}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-1} + r_{\varepsilon, \lambda}^{3} r^{-2} \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-2}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-2} + r_{\varepsilon, \lambda}^{3} r^{-3} \biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{1} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{ S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}( B_{\frac{r_{0}}{2}}(\tilde{x}^{1}))} \leq \sup_{r_{\varepsilon, \lambda} \leq r \leq {\frac{r_{0}}{2}}} r^{4 - \nu} \bigl\vert \tilde{ S_{1}}(0,0) \bigr\vert \leq c_{\kappa } \varepsilon ^{4} r_{\varepsilon, \lambda}^{-4} + c_{\kappa }\lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}}(\tilde{x}^{1})- B_{\frac{r_{0}}{2}}(\tilde{x}^{1})\), using the estimate (96), we have
$$\begin{aligned} &\bigl\vert \tilde{S}_{1}(0,0) \bigr\vert \\ &\quad \leq c_{\kappa} \varepsilon ^{4} r^{-8 (1 + \eta _{1})} + \biggl\vert \bigl[\Delta ^{2},\chi _{r_{0}} \bigl(x - \tilde{x}^{1} \bigr) \bigr]H^{\mathrm{ext}} \biggl(\tilde{ \varphi}^{1}_{1}, \tilde{\psi}_{1}^{1}; \frac{x - \tilde{x}^{1}}{r_{\varepsilon, \lambda}} \biggr) \biggr\vert \\ &\qquad{} +\lambda \Biggl\vert \Biggl(\Delta \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr)^{2} \Biggr\vert \\ & \qquad{}+ \lambda \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \\ &\qquad{}\times \nabla \Biggl(\Delta \Biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{ \tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr) \Biggr\vert \\ &\qquad{} + \lambda ^{2} \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert ^{2} \\ & \qquad{}\times \Biggl\vert \Delta \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert \\ &\quad\leq c_{\kappa} \bigl( \varepsilon ^{4} r^{-8 (1 + \eta _{1})} + r^{-1} r_{\varepsilon, \lambda}^{3} \bigr) + c_{\kappa} \lambda \biggl\vert \biggl( \frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-2}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-2} + r_{ \varepsilon, \lambda}^{3} r^{-3} \biggr)^{2} \biggr\vert \\ &\qquad{}+ c_{\kappa} \lambda \biggl\vert \biggl(\frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-1}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-1} + r_{\varepsilon, \lambda}^{3} r^{-2} \biggr) \biggl(\frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-3} \\ &\qquad{}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-3} + r_{ \varepsilon, \lambda}^{3} r^{-4} \biggr) \biggr\vert \\ &\qquad{} + c_{\kappa} \lambda ^{2} \biggl\vert \frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-1}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-1} + r_{\varepsilon, \lambda}^{3} r^{-2} \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \frac{1 + \eta _{1}}{\gamma} \bigl\vert x- \tilde {x}^{1} \bigr\vert ^{-2}+ (1 + \eta _{2}) \bigl\vert x- \tilde {x}^{2} \bigr\vert ^{-2} + r_{\varepsilon, \lambda}^{3} r^{-3} \biggr\vert , \end{aligned}$$
where
$$\begin{aligned} \bigl[\Delta ^{2}, \chi _{r_{0}} \bigr] w ={}& w \Delta ^{2} \chi _{r_{0}} + 2 \Delta w \Delta \chi _{r_{0}} + 4 \nabla (\Delta w)\cdot \nabla \chi _{r_{0}} + 4 \nabla w \cdot \nabla ( \Delta \chi _{r_{0}})\\ &{}+4 \sum_{i,j=1}^{4}{ \frac{\partial ^{2}\chi _{r_{0}}}{\partial x_{i} \partial x_{j}}} {\frac{\partial ^{2}w}{\partial x_{i}\partial x_{j}}}. \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{1} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}(B_{r_{0}}( \tilde{x}^{1})-B_{{\frac{r_{0}}{2}}}(\tilde{x}^{1}))} \leq \sup_{{\frac{r_{0}}{2}} \leq r \leq r_{0}} r^{4 - \nu} \bigl\vert \tilde{ S_{1}}(0,0) \bigr\vert \leq c_{\kappa } r_{\varepsilon, \lambda}^{2} + c_{\kappa } \lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}/2}(\tilde{x}^{2}) - B_{r_{\varepsilon, \lambda}}(\tilde{x}^{2})\), we have \(\chi _{r_{0}}(x-\tilde{x}^{1})=0, \chi _{r_{0}}(x-\tilde{x}^{2})=1, \chi _{r_{0}}(x-\tilde{x}^{3})=0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{1}}}= 0\), so that \(\tilde{S_{1}}(0,0) = \rho ^{4} e^{\gamma \tilde{\mathbf {w}_{1}}+\mathbf{ (1 }-\boldsymbol{ \gamma ) }\tilde{\mathbf {w}_{2}}} - \mathcal{L}_{\lambda}( \tilde{\mathbf{w}}_{1})\). Then
$$\begin{aligned} & \bigl\vert \tilde{S_{1}}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa}\varepsilon ^{4} \bigl\vert x- \tilde{x}^{2} \bigr\vert ^{-8 (1+\eta _{2})} \\ &\qquad{}+\lambda \biggl\vert \biggl( \Delta \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{2}, \tilde {\psi}_{1}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr)^{2} \biggr\vert \\ & \qquad{}+ \lambda \biggl\vert \nabla \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{2}, \tilde {\psi}_{1}^{2}} \biggl({ \frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \\ &\qquad{}\cdot \nabla \biggl(\Delta \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{2}, \tilde {\psi}_{1}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr) \biggr\vert \\ &\qquad{}+ \lambda ^{2} \biggl\vert \nabla \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{2}, \tilde {\psi}_{1}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \Delta \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{2}, \tilde {\psi}_{1}^{2}} \biggl({ \frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{2} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu -4}( B_{r_{0}}( \tilde{x}^{2}))} \leq \sup_{r_{\varepsilon, \lambda}\leq r \leq r_{0}/2} r^{4-\nu} \bigl\vert \tilde{S_{1}}(0,0) \bigr\vert \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{\kappa }\lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}}(\tilde{x}^{2})- B_{r_{0}/2}(\tilde{x}^{2})\), using the estimate (96), there holds
$$\begin{aligned} &\bigl\vert \tilde{S_{1}}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa} \varepsilon ^{4} r^{-8 (1+\eta _{2})} + \biggl\vert \bigl[\Delta ^{2},\chi _{r_{0}} \bigl(x- \tilde{x}^{2} \bigr) \bigr] H^{\mathrm{ext}} \biggl(\tilde{ \varphi}^{2}_{1}, \tilde{\psi}_{1}^{2}; \frac{x - \tilde{x}^{2}}{r_{\varepsilon, \lambda}} \biggr) \biggr\vert \\ &\qquad{} +\lambda \Biggl\vert \Biggl(\Delta \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr)^{2} \Biggr\vert \\ &\qquad{} + \lambda \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \\ &\qquad{}\times \nabla \Biggl(\Delta \Biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{ \tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr) \Biggr\vert \\ & \qquad{}+ \lambda ^{2} \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert ^{2} \\ & \qquad{}\times \Biggl\vert \Delta \Biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{1}^{i}, \tilde {\psi}_{1}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{2} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu -4}( B_{r_{0}}( \tilde{x}^{2}))} \leq \sup_{r_{0}/2 \leq r \leq r_{0}} r^{4-\nu} \bigl\vert \tilde{S_{1}}(0,0) \bigr\vert \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{ \kappa }\lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
Similarly, for \(\nu \in (-1,0) \) and \(\eta _{3} \) small enough, we can prove the same result for \(\tilde{x}^{3}\).
• In \(\Omega - ( B_{r_{0}}(\tilde{x}^{1}) \cup B_{r_{0}}(\tilde{x}^{2}) \cup B_{r_{0}}(\tilde{x}^{3}) )\), we have \(\chi _{r_{0}}(x - \tilde{x}^{1}) = 0, \chi _{r_{0}}(x - \tilde{x}^{2}) = 0, \chi _{r_{0}}(x - \tilde{x}^{3}) = 0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{1}}} = 0\). Then
$$\begin{aligned} &\bigl\vert \tilde{S_{1}}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa}\varepsilon ^{4} +\lambda \biggl\vert \biggl(\Delta \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + \lambda \biggl\vert \nabla \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) \biggr) \\ &\qquad{}\times \nabla \biggl(\Delta \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) \biggr) \biggr) \biggr\vert \\ &\qquad{}+ \lambda ^{2} \biggl\vert \nabla \biggl(\frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) \biggr) \biggr\vert ^{2} \\ &\qquad{}\times \biggl\vert \Delta \biggl( \frac{1 + \eta _{1}}{\gamma} G \bigl(x,\tilde {x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \biggr) \biggr\vert . \end{aligned}$$
So for \(\nu \in (-1, 0)\), we have
$$\begin{aligned} \bigl\Vert \tilde{ S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}( \bar{\Omega} - \bigcup_{i=1}^{3} B_{r_{0}}(\tilde{x}^{i}) )} \leq \sup_{ r \geq r_{0} } r^{4 - \nu} \bigl\vert \tilde{ S_{1}}(0,0) \bigr\vert \leq c_{\kappa } \varepsilon ^{4} + c_{\kappa }\lambda. \end{aligned}$$
We conclude that
$$\begin{aligned} \bigl\Vert \tilde{ S_{1}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4} ( \bar{\Omega}_{r_{0}}({\tilde{\mathbf {x}}}))} \leq c_{\kappa } r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
(97)
Now, we are interested in the second equation of the previous system.
• In \(B_{{\frac{r_{0}}{2}}}(\tilde{x}^{1})- B_{r_{\varepsilon, \lambda}}( \tilde{x}^{1})\), we have \(\chi _{r_{0}}(x - \tilde{x}^{1}) = 1, \chi _{r_{0}}(x - \tilde{x}^{2}) = 0, \chi _{r_{0}}(x - \tilde{x}^{3}) = 0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{1}}}= 0\), so that \(|\tilde{S}_{2}(0,0)| = \rho ^{4} e^{ \xi \tilde{w}_{2} + (1-\xi ) \tilde{w}_{1}}- \mathcal{L}_{\lambda}(\tilde{\mathbf{w}}_{2})\). Then
$$\begin{aligned} & \bigl\vert \tilde{S}_{2}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa} \varepsilon ^{4} \bigl\vert x - \tilde{x}^{1} \bigr\vert ^{-8 \frac{(1-\xi )(1 + \eta _{1})}{\gamma}} \\ &\qquad{}+ \lambda \biggl\vert \biggl(\Delta \biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{1}, \tilde {\psi}_{2}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + \lambda \biggl\vert \nabla \biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{1}, \tilde {\psi}_{2}^{1}} \biggl({ \frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \\ &\qquad{}\times \nabla \biggl(\Delta \biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{1}, \tilde {\psi}_{2}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr) \biggr\vert + \lambda ^{2} \biggl\vert \nabla \biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{1}, \tilde {\psi}_{2}^{1}} \biggl({\frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \Delta \biggl(\frac{1 + \eta _{3}}{\gamma} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{1}, \tilde {\psi}_{2}^{1}} \biggl({ \frac{x - \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{1} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{ S_{2}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}( B_{\frac{r_{0}}{2}}(\tilde{x}^{1}))} \leq \sup_{r_{\varepsilon, \lambda} \leq r \leq {\frac{r_{0}}{2}}} r^{4 - \nu} \bigl\vert \tilde{ S_{2}}(0,0) \bigr\vert \leq c_{\kappa } r_{\varepsilon, \lambda}^{2} + c_{\kappa } \lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}}(\tilde{x}^{1})- B_{\frac{r_{0}}{2}}(\tilde{x}^{1})\), using the estimate (96), we have
$$\begin{aligned} &\bigl\vert \tilde{S}_{2}(0,0) \bigr\vert \\ &\quad \leq c_{\kappa} \varepsilon ^{4} r^{-8 \frac{(1-\xi )(1 + \eta _{1})}{\gamma}} + \biggl\vert \bigl[ \Delta ^{2},\chi _{r_{0}} \bigl(x - \tilde{x}^{1} \bigr) \bigr]H^{\mathrm{ext}} \biggl( \tilde{ \varphi}^{1}_{2}, \tilde{\psi}_{2}^{1}; \frac{x - \tilde{x}^{1}}{r_{\varepsilon, \lambda}} \biggr) \biggr\vert \\ &\qquad{} +\lambda \Biggl\vert \Biggl(\Delta \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr)^{2} \Biggr\vert \\ &\qquad{} + \lambda \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr) + (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \\ &\qquad{}\times \nabla \Biggl(\Delta \Biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{ \tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr) \Biggr\vert \\ & \qquad{}+ \lambda ^{2} \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert ^{2} \\ & \qquad{}\times \Biggl\vert \Delta \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert , \end{aligned}$$
where
$$\begin{aligned} \bigl[\Delta ^{2}, \chi _{r_{0}} \bigr] w = {}&w \Delta ^{2} \chi _{r_{0}} + 2 \Delta w \Delta \chi _{r_{0}} + 4 \nabla (\Delta w)\cdot \nabla \chi _{r_{0}} + 4 \nabla w \cdot \nabla ( \Delta \chi _{r_{0}})\\ &{}+4 \sum_{i,j=1}^{4}{ \frac{\partial ^{2}\chi _{r_{0}}}{\partial x_{i} \partial x_{j}}} {\frac{\partial ^{2}w}{\partial x_{i}\partial x_{j}}}. \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{1} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{2}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}(B_{r_{0}}( \tilde{x}^{1})-B_{{\frac{r_{0}}{2}}}(\tilde{x}^{1}))} \leq \sup_{{\frac{r_{0}}{2}} \leq r \leq r_{0}} r^{4 - \nu} \bigl\vert \tilde{ S_{2}}(0,0) \bigr\vert \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}+ c_{\kappa } \lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}/2}(\tilde{x}^{2}) - B_{r_{\varepsilon, \lambda}}(\tilde{x}^{2})\), we have \(\chi _{r_{0}}(x-\tilde{x}^{1})=0, \chi _{r_{0}}(x-\tilde{x}^{2})=1, \chi _{r_{0}}(x-\tilde{x}^{3})=0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{2}}}= 0\), so that \(\tilde{S_{2}}(0,0) = \rho ^{4} e^{\xi \tilde{\mathbf {w}_{2}}+\mathbf{ (1 }-\boldsymbol{ \xi ) }\tilde{\mathbf {w}_{1}}}\). Then
$$\begin{aligned} & \bigl\vert \tilde{S_{2}}(0,0) \bigr\vert \\ &\quad \leq c_{\kappa}\varepsilon ^{4} \bigl\vert x- \tilde{x}^{2} \bigr\vert ^{-8 (1+\eta _{2})}\\ &\qquad{} +\lambda \biggl\vert \biggl( \Delta \biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr) + (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{2}, \tilde {\psi}_{2}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr)^{2} \biggr\vert \\ &\qquad{} + \lambda \biggl\vert \nabla \biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{2}, \tilde {\psi}_{2}^{2}} \biggl({ \frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \\ &\qquad{}\times \nabla \biggl(\Delta \biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) \\ &\qquad{}+ H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{2}, \tilde {\psi}_{2}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr) \biggr\vert \\ &\qquad{}+ \lambda ^{2} \biggl\vert \nabla \biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{2}, \tilde {\psi}_{2}^{2}} \biggl({\frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert ^{2} \\ &\qquad{} \times \biggl\vert \Delta \biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{2}, \tilde {\psi}_{2}^{2}} \biggl({ \frac{x - \tilde {x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \biggr) \biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{2} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{2}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu -4}( B_{r_{0}}( \tilde{x}^{2}))} \leq \sup_{r_{\varepsilon, \lambda}\leq r \leq r_{0}/2} r^{4-\nu} \bigl\vert \tilde{S_{2}}(0,0) \bigr\vert \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{\kappa }\lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
• In \(B_{r_{0}}(\tilde{x}^{2})- B_{r_{0}/2}(\tilde{x}^{2})\), using the estimate (96), there holds
$$\begin{aligned} &\bigl\vert \tilde{S_{2}}(0,0) \bigr\vert \\ &\quad\leq c_{\kappa} \varepsilon ^{4} r^{-8 (1+\eta _{2})} + \biggl\vert \bigl[\Delta ^{2},\chi _{r_{0}} \bigl(x- \tilde{x}^{2} \bigr) \bigr]H^{\mathrm{ext}} \biggl(\tilde{ \varphi}^{2}_{2}, \tilde{\psi}_{2}^{2}; \frac{x - \tilde{x}^{2}}{r_{\varepsilon, \lambda}} \biggr) \biggr\vert \\ & \qquad{}+\lambda \Biggl\vert \Biggl(\Delta \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr)^{2} \Biggr\vert \\ &\qquad{} + \lambda \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \\ &\qquad{}\times \nabla \Biggl(\Delta \Biggl( \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x,\tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{ \tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr) \Biggr\vert \\ &\qquad{} + \lambda ^{2} \Biggl\vert \nabla \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert ^{2} \\ & \qquad{}\times \Biggl\vert \Delta \Biggl(\frac{1 + \eta _{3}}{\xi} G \bigl(x,\tilde {x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde {x}^{2} \bigr) + \sum_{i = 1}^{3} \chi _{r_{0}} \bigl(x- \tilde {x}^{i} \bigr) H^{\mathrm{ext}}_{\tilde {\varphi}_{2}^{i}, \tilde {\psi}_{2}^{i}} \biggl({\frac{x - \tilde {x}^{i}}{r_{\varepsilon, \lambda}}} \biggr) \Biggr) \Biggr\vert . \end{aligned}$$
Hence, for \(\nu \in (-1,0)\) and \(\eta _{2} \) small enough, we get
$$\begin{aligned} \bigl\Vert \tilde{S_{2}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu -4}( B_{r_{0}}( \tilde{x}^{2}))} \leq \sup_{r_{0}/2 \leq r \leq r_{0}} r^{4-\nu} \bigl\vert \tilde{S_{2}}(0,0) \bigr\vert \leq c_{\kappa }r_{\varepsilon, \lambda}^{2}+ c_{ \kappa }\lambda + c_{\kappa }\lambda r_{\varepsilon, \lambda}^{3}. \end{aligned}$$
Similarly, for \(\nu \in (-1,0) \) and \(\eta _{3} \) small enough, we can prove the same result for \(\tilde{x}^{3}\).
• In \(\Omega - ( B_{r_{0}}(\tilde{x}^{1}) \cup B_{r_{0}}(\tilde{x}^{2}) \cup B_{r_{0}}(\tilde{x}^{3}) )\), we have \(\chi _{r_{0}}(x - \tilde{x}^{1}) = 0, \chi _{r_{0}}(x - \tilde{x}^{2}) = 0, \chi _{r_{0}}(x - \tilde{x}^{3}) = 0\) and \(\Delta ^{2} {\tilde{\mathbf {w}_{1}}} = 0\). So for \(\nu \in (-1, 0)\), we have
$$\begin{aligned} \bigl\Vert \tilde{ S_{2}}(0,0) \bigr\Vert _{{\mathcal {C}}^{0, \alpha}_{\nu - 4}( \bar{\Omega} - \bigcup_{i=1}^{3} B_{r_{0}}(\tilde{x}^{i}) )} \leq \sup_{ r \geq r_{0} } r^{4 - \nu} \bigl\vert \tilde{ S_{2}}(0,0) \bigr\vert \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
Making use of Proposition 3 together with (91), we conclude that
$$\begin{aligned} \bigl\Vert \tilde{\mathcal{N}} (0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})) } \leq c_{\kappa } r_{ \varepsilon, \lambda}^{2}\quad \text{and}\quad \bigl\Vert \tilde{\mathcal{M}} (0,0) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) } \leq c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
(98)
For the proof of the third estimate, let \(\tilde{v}_{1}, \tilde{v}_{2}, \tilde{v}_{1}'\) and \(\tilde{v}_{2}' \in {\mathcal {C}}^{4, \alpha}_{\nu}(\bar{\Omega}^{*})\) satisfy (95), we have
$$\begin{aligned} & \bigl\vert \tilde{ S_{1} }(\tilde{v}_{1}, \tilde{v}_{2}) - \tilde { S_{1}} \bigl( \tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\vert \\ &\quad \leq c_{\kappa} \varepsilon ^{4} e^{\gamma \tilde{\mathbf{w}}_{1} + (1-\gamma ) \tilde{\mathbf{w}}_{2}} \bigl\vert e^{\gamma \tilde{v}_{1} + (1- \gamma ) \tilde{v}_{2}} - e^{\gamma \tilde{v}_{1}' + (1-\gamma ) \tilde{v}_{2}'} \bigr\vert + \bigl\vert \mathcal{L}_{\lambda} ( \tilde{\mathbf{w}}_{1} + \tilde{v}_{1} ) - \mathcal{L}_{\lambda} \bigl(\tilde{ \mathbf{w}}_{1} + \tilde{v}_{1}' \bigr) \bigr\vert \\ &\quad \leq c_{\kappa} \varepsilon ^{4} \bigl( \gamma \bigl\vert \tilde{v}_{1} - \tilde{v}_{1}' \bigr\vert + (1- \gamma ) \bigl\vert \tilde{v}_{2} - \tilde{v}_{2}' \bigr\vert \bigr)+ \lambda \bigl\vert \Delta \bigl(\tilde{v}_{1}- \tilde{v}_{1}' \bigr) \bigr\vert \bigl(2 \vert \Delta \tilde{\mathbf{w}}_{1} \vert + \vert \Delta \tilde{v}_{1} \vert + \bigl\vert \Delta \tilde{v}_{1}' \bigr\vert \bigr) \\ &\qquad{} +\lambda \vert \nabla \bigl(\tilde{v}_{1}-\tilde{v}_{1}' \bigr) \cdot \nabla \bigl(2\Delta \tilde{ \mathbf{w}}_{1} + \Delta \tilde{v}_{1} + \Delta \tilde{v}_{1}' \bigr) +\nabla (\Delta \bigl(\tilde{v}_{1}-\tilde{v}_{1}' \bigr) \cdot \nabla \bigl(2 \tilde{\mathbf{w}}_{1} + \tilde{v}_{1} + \tilde{v}_{1}' \bigr) \vert \\ &\qquad{}+\lambda ^{2} \vert \Delta \bigl(\tilde{v}_{1}-\tilde{v}_{1}' \bigr) \vert \bigl[ \bigl\vert \nabla (\tilde{ \mathbf{w}}_{1} +\tilde{v}_{1} ) \bigr\vert ^{2} + \bigl\vert \nabla \bigl(\tilde{\mathbf{w}}_{1} + \tilde{v}_{1}' \bigr) \bigr\vert ^{2} \bigr] \\ &\qquad{} + \lambda ^{2} \bigl( \vert \Delta \tilde{\mathbf{w}}_{1} \vert + \vert \Delta \tilde{v}_{1} \vert + \bigl\vert \Delta \tilde{v}_{1}' \bigr\vert \bigr) \bigl[ \bigl\vert \nabla (\tilde{\mathbf{w}}_{1} +\tilde{v}_{1} ) \bigr\vert ^{2} - \bigl\vert \nabla \bigl(\tilde{\mathbf{w}}_{1} + \tilde{v}_{1}' \bigr) \bigr\vert ^{2} \bigr]. \end{aligned}$$
So, for \(\eta _{i} \), \(i=1,2,3\), small enough and using the fact that for all \(w \in {\mathcal {C}}^{4, \alpha}_{\nu}(\bar{\Omega}^{*})\), there exists a constant \(c>0\) such that \(\vert \nabla ^{l} w\vert \leq c_{\kappa }r^{\nu -l}\| w \|_{{ \mathcal {C}}^{4, \alpha}_{\nu }(\bar{\Omega}^{*}({\tilde{\mathbf {x}}})) }\), we get
$$\begin{aligned} &\sup_{x \in \bar{\Omega}^{*}} r^{4-\nu} \bigl\vert \tilde{ S_{1} }(\tilde{v}_{1}, \tilde{v}_{2}) - \tilde { S_{1}} \bigl( \tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\vert \\ &\quad \leq c_{\kappa} \varepsilon ^{4} \sup_{x \in \bar{\Omega}^{*}} r^{4-\nu} \bigl( \gamma r^{\nu } \bigl\Vert \tilde{v}_{1} - \tilde{v}_{1}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )} \\ &\qquad{}+ (1-\gamma )r^{\nu} \bigl\Vert \tilde{v}_{2} - \tilde{v}_{2}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )} \bigr) + c_{\kappa} \lambda \bigl\Vert \tilde{v}_{1} - \tilde{v}_{1}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )}. \end{aligned}$$
Using the estimate (91), there exists \({c}_{\kappa} \) (depending on κ) such that
$$\begin{aligned} &\bigl\Vert \tilde{\mathcal {N}}(\tilde{v}_{1}, \tilde{v}_{2}) - \tilde {\mathcal {N}} \bigl(\tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\nu} (\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))} \\ &\quad\leq {c}_{\kappa} r_{\varepsilon,\lambda}^{2} \bigl( \bigl\Vert \tilde{v}_{1} - \tilde{v}_{1}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({ \tilde{\mathbf {x}}}) )} + \bigl\Vert \tilde{v}_{2} - \tilde{v}_{2}' \bigr\Vert _{C^{4,\alpha}_{ \nu}( \bar{\Omega}^{*}({\tilde{\mathbf {x}}}) )} \bigr). \end{aligned}$$
(99)
Similarly, we can use the same arguments to prove,
$$\begin{aligned} & \bigl\Vert \tilde{\mathcal {M} }(\tilde{v}_{1}, \tilde{v}_{2}) - \tilde {\mathcal {M} } \bigl(\tilde{v}_{1}', \tilde{v}_{2}' \bigr) \bigr\Vert _{\mathcal{C}^{4, \alpha}_{\nu} (\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))} \\ &\quad \leq {c}_{\kappa} r_{\varepsilon,\lambda}^{2} \bigl( \bigl\Vert \tilde{v}_{1} - \tilde{v}_{1}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({\tilde{\mathbf {x}}}))} + \bigl\Vert \tilde{v}_{2} - \tilde{v}_{2}' \bigr\Vert _{C^{4, \alpha}_{\nu}(\bar{\Omega}^{*}({ \tilde{\mathbf {x}}}))} \bigr). \end{aligned}$$
(100)
□
Reducing \(\varepsilon _{\kappa}\) and \(\lambda _{\kappa }>0\), if necessary, we can assume that \({c}_{\kappa} r_{\varepsilon, \lambda}^{2} \leq \frac{1}{2}\) for all \(\varepsilon \in (0, \varepsilon _{\kappa} )\) and \(\lambda \in (0, \lambda _{\kappa})\). Then, (99) and (100) are enough to show that
$$\begin{aligned} (\tilde{v}_{1}, \tilde{v}_{2}) \mapsto \bigl( \tilde{ \mathcal{N} }( \tilde{v}_{1}, \tilde{v}_{2}) , \tilde{ \mathcal{M} }(\tilde{v}_{1}, \tilde{v}_{2}) \bigr) \end{aligned}$$
is a contraction from the ball
$$\begin{aligned} \bigl\{ (\tilde{v}_{1}, \tilde{v}_{2}) \in \bigl( C^{4, \alpha}_{\nu} \bigl( \mathbb{R}^{4} \bigr) \bigr)^{2}: \bigl\Vert (\tilde{v}_{1}, \tilde{v}_{2}) \bigr\Vert _{( C^{4, \alpha}_{\nu}(\mathbb{R}^{4}))^{2}} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2} \bigr\} , \end{aligned}$$
into itself. Hence there exists a unique fixed point \((\tilde{v}_{1},\tilde{v}_{2}) \) in this set, which is a solution of (92). Applying a fixed point theorem for contraction mappings, we conclude that
Proposition 10
Given \(\kappa >0\), there exists \(\varepsilon _{\kappa }>0\), \(\lambda _{\kappa }>0\) (depending on κ) such that for any \(\varepsilon \in (0, \varepsilon _{\kappa})\), \(\lambda \in (0, \lambda _{\kappa})\), \(\eta _{i}\) and \(\tilde{x}^{i}\) satisfying (94) and functions \(\tilde{\varphi}^{i}_{j}\) and \(\tilde{\psi}^{i}_{j}\) satisfying (66) and (93), there exists a unique \((\tilde{v}_{1}, \tilde{v}_{2})(:= (\tilde{v}_{1,\varepsilon, \eta _{1}, \eta _{2}, {\tilde{\mathbf {x}}}, \tilde{\varphi}^{i}_{1}, \tilde{\psi}^{i}_{1}}, \tilde{v}_{2,\varepsilon, \eta _{2},\eta _{3}, {\tilde{\mathbf {x}}}, \tilde{\varphi}^{i}_{2}, \tilde{\psi}^{i}_{2}}))\) solution of (92) so that for \(v_{k} (k = 1, 2)\) defined by
$$\begin{aligned} &v_{1} (x):= \frac{1 + \eta _{1}}{\gamma} G \bigl(x, \tilde{x}^{1} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde{x}^{2} \bigr) + \sum_{i=1}^{3} \chi _{r_{0}} \bigl(x - \tilde {x}^{i} \bigr) H^{\mathrm{ext}} \biggl( \tilde {\varphi}^{i}_{1},\tilde { \psi}^{i}_{1}; \frac{x-\tilde {x}^{i}}{ r_{\varepsilon, \lambda}} \biggr) + \tilde {v}_{1}(x), \\ & v_{2} (x):= \frac{1 + \eta _{3}}{\xi} G \bigl(x, \tilde{x}^{3} \bigr)+ (1 + \eta _{2}) G \bigl(x, \tilde{x}^{2} \bigr) + \sum _{i=1}^{3} \chi _{r_{0}} \bigl(x - \tilde {x}^{i} \bigr) H^{\mathrm{ext}} \biggl(\tilde { \varphi}^{i}_{2}, \tilde {\psi}^{i}_{2}; \frac{x-\tilde {x}^{i}}{ r_{\varepsilon, \lambda}} \biggr) + \tilde {v}_{2}(x) \end{aligned}$$
solves (89) in \(\bar{\Omega}_{r_{\varepsilon, \lambda}}({\tilde{\mathbf {x}}})\). In addition, we have
$$\begin{aligned} \bigl\Vert (\tilde {v}_{1}, \tilde {v}_{2}) \bigr\Vert _{C^{4,\alpha}_{\nu}( \bar{\Omega}^{*}({\tilde{\mathbf {x}}}))} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
3.4 The nonlinear Cauchy-data matching
We will summarize the results of the previous sections. Using the previous notations, assume that \({\tilde{\mathbf {x}}}: = (\tilde {x}^{1}, \tilde {x}^{2}, \tilde {x}^{3}) \in \Omega ^{3}\) are given close to \({\mathbf{x}}: = (x^{1}, x^{2}, x^{3})\). Assume also that
$$\begin{aligned} {\boldsymbol{\tau}}: = (\tau _{1}, \tau _{2}, \tau _{3}) \in \bigl[\tau _{1}^{-} ,\tau _{1}^{+} \bigr]\times \bigl[\tau _{2}^{-}, \tau _{2}^{+} \bigr]\times \bigl[\tau _{3}^{-} ,\tau _{3}^{+} \bigr]\subset (0, \infty )^{3}, \end{aligned}$$
are given (the values of \(\tau _{l}^{-}\) and \(\tau _{l}^{+}\), for \(l=1,2,3\) will be fixed later). First, we consider some set of boundary data \({\boldsymbol{\varphi}}^{i}: = (\varphi ^{i}_{1}, \varphi ^{i}_{2}) \in ({ \mathcal{C}}^{4,\alpha}(S^{3}))^{2}\) and \({\boldsymbol{\psi}}^{i}: = (\psi ^{i}_{1}, \psi ^{i}_{2}) \in ({\mathcal{C}}^{2, \alpha}(S^{3}))^{2}\). Let \(\varepsilon \in (0, \varepsilon _{\kappa})\) and according to the result of Proposition 7, 8, and 9, we can find, \(u_{\mathrm{int}}:=( u_{\mathrm{int},1}, u_{\mathrm{int},2})\) a solution of (39) in \(B_{r_{\varepsilon, \lambda}}(\tilde {x}^{1}) \cup B_{r_{ \varepsilon, \lambda}}(\tilde {x}^{2})\cup B_{r_{\varepsilon, \lambda}}(\tilde {x}^{3})\), which can be decomposed as
$$\begin{aligned} u_{\mathrm{int},1} (x):= \textstyle\begin{cases} {\frac{1}{\gamma}}u_{\varepsilon,\tau _{1}}(x-\tilde{x}^{1}) - {\frac{1- \gamma}{\gamma}} G(x, \tilde {x}^{2})- {\frac{1-\gamma}{\gamma \xi}} G(x, \tilde {x}^{3})- {\frac{\ln \gamma}{\gamma}} + H^{\mathrm{int}}_{ \varphi ^{1}_{1},\psi _{1}^{1}}( \frac{x-\tilde{x}^{1}}{r_{\varepsilon, \lambda}}) \\ \quad{}+ h_{1}^{1} ( \frac{R^{1}_{\varepsilon, \lambda}(x-\tilde{x}^{1})}{r_{\varepsilon, \lambda}} )+ v_{1}^{1} ( \frac{R^{1}_{\varepsilon, \lambda}(x-\tilde{x}^{1})}{r_{\varepsilon, \lambda}} )\\ \qquad \text{in } B_{r_{\varepsilon, \lambda}}(\tilde{ x}^{1}), \\ u_{\varepsilon,\tau _{2}}(x-\tilde{x}^{2}) + H^{\mathrm{int}}_{ \varphi ^{2}_{1},\psi _{1}^{2}}( \frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}) + h_{1}^{2} ( \frac{R^{2}_{\varepsilon, \lambda}(x-\tilde{x}^{2})}{r_{\varepsilon, \lambda}} )+ v_{1}^{2} ( \frac{R^{2}_{\varepsilon, \lambda}(x-\tilde{x}^{2})}{r_{\varepsilon, \lambda}} )\\ \qquad \text{in } B_{r_{\varepsilon, \lambda}}(\tilde{ x}^{2}), \\ {\frac{1}{\gamma}}G(x, \tilde {x}^{1}) +G(x, \tilde {x}^{2})+ H^{\mathrm{int}}_{\varphi ^{3}_{1},\psi _{1}^{3}}( \frac{x-\tilde{x}^{3}}{r_{\varepsilon, \lambda}}) + h_{1}^{3} ( \frac{R^{3}_{\varepsilon, \lambda}(x-\tilde{x}^{3})}{r_{\varepsilon, \lambda}} )+ v_{1}^{3} ( \frac{R^{3}_{\varepsilon, \lambda}(x-\tilde{x}^{3})}{r_{\varepsilon, \lambda}} )\\ \qquad\text{in } B_{r_{\varepsilon, \lambda}}(\tilde {x}^{3}) \end{cases}\displaystyle \end{aligned}$$
and
$$\begin{aligned} u_{\mathrm{int},2}(x):= \textstyle\begin{cases} {\frac{1}{\xi}} G(x, \tilde {x}^{3})+G(x, \tilde {x}^{2}) + H^{\mathrm{int}}_{\varphi ^{1}_{2},\psi _{2}^{1}}( \frac{x-\tilde{x}^{1}}{r_{\varepsilon, \lambda}}) + h_{2}^{1} ( \frac{R^{1}_{\varepsilon, \lambda}(x-\tilde{x}^{1})}{r_{\varepsilon, \lambda}} )+ v_{2}^{1} ( \frac{R^{1}_{\varepsilon, \lambda}(x-\tilde{x}^{1})}{r_{\varepsilon, \lambda}} ) \\ \qquad\text{in } B_{r_{\varepsilon, \lambda}}(\tilde {x}^{1}), \\ u_{\varepsilon,\tau _{2}}(x-\tilde{x}^{2}) + H^{\mathrm{int}}_{ \varphi ^{2}_{2},\psi _{2}^{2}} ( \frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}) + h_{2}^{2} ( \frac{R^{2}_{\varepsilon, \lambda}(x-\tilde{x}^{2})}{r_{\varepsilon, \lambda}} )+ v_{2}^{2} ( \frac{R^{2}_{\varepsilon, \lambda}(x-\tilde{x}^{2})}{r_{\varepsilon, \lambda}} )\\ \qquad \text{in } B_{r_{\varepsilon, \lambda}}(\tilde{ x}^{2}), \\ {\frac{1}{\xi}} u_{\varepsilon,\tau _{3}}(x-\tilde{x}^{3}) - {\frac{1-\xi}{\xi}} G(x, \tilde {x}^{2})-{\frac{1-\xi}{\gamma \xi}} G(x, \tilde {x}^{1})-{\frac{\ln \xi}{\xi}} + H^{\mathrm{int}}_{\varphi ^{3}_{2}, \psi _{2}^{3}}(\frac{x-\tilde{x}^{3}}{r_{\varepsilon, \lambda}}) \\ \quad{}+ h_{2}^{3} ( \frac{R^{3}_{\varepsilon, \lambda}(x-\tilde{x}^{3})}{r_{\varepsilon, \lambda}} )+ v_{2}^{3} ( \frac{R^{3}_{\varepsilon, \lambda}(x-\tilde{x}^{3})}{r_{\varepsilon, \lambda}} ) \\ \qquad \text{in } B_{r_{\varepsilon, \lambda}}(\tilde{ x}^{3}), \end{cases}\displaystyle \end{aligned}$$
where for \(i\in \{1,2,3\}\) and \(j\in \{1,2\}\), \(R_{\varepsilon, \lambda}^{i}=\tau _{i}{\frac{r_{ \varepsilon, \lambda}}{\varepsilon}}\) and the functions \(h_{j}^{i}\) satisfy
$$\begin{aligned} &\bigl\Vert \bigl(h_{1}^{1}, h_{2}^{1} \bigr) \bigr\Vert _{{\mathcal {C}}^{4, \alpha}_{\mu }( \mathbb{R}^{4})\times{\mathcal {C}}^{4, \alpha}_{\delta }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2},\qquad \bigl\Vert \bigl(h_{1}^{2}, h_{2}^{2} \bigr) \bigr\Vert _{({\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4}))^{2}} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2} \quad\text{and} \\ &\bigl\Vert \bigl(h_{1}^{3}, h_{2}^{3} \bigr) \bigr\Vert _{({\mathcal {C}}^{4, \alpha}_{\delta }( \mathbb{R}^{4})\times {\mathcal {C}}^{4, \alpha}_{\mu }(\mathbb{R}^{4})} \leq 2 c_{\kappa } r_{\varepsilon, \lambda}^{2}. \end{aligned}$$
Similarly, given some boundary data \({ \tilde{\varphi}_{j}^{i}} \in C^{4, \alpha}(S^{3}) \), \({ \tilde{\psi}_{j}^{i}} \in C^{2, \alpha}(S^{3}) \) satisfying (66), \((\eta _{1}, \eta _{2}, \eta _{3}) \in \mathbb{R}^{3} \) satisfying (94), provided \(\varepsilon \in (0, \varepsilon _{\kappa})\), by Proposition 10, we find a solution \(u_{\mathrm{ext}}:=(u_{\mathrm{ext},1},u_{\mathrm{ext},2})\) of (39) in \(\bar{\Omega} \setminus ( B_{r_{\varepsilon, \lambda}} (\tilde {x}^{1}) \cup B_{r_{\varepsilon, \lambda}} (\tilde {x}^{2}))\cup B_{r_{ \varepsilon, \lambda}} (\tilde {x}^{3}))\), which can be decomposed as
$$\begin{aligned} \textstyle\begin{cases} u_{\mathrm{ext},1} (x):= \frac{1+ \eta _{1}}{\gamma} G(x,\tilde{ x}^{1})+ (1+ \eta _{2}) G(x, \tilde{x}^{2}) + \sum_{i=1}^{3}\chi _{r_{0}} (x- \tilde{ x}^{i}) H^{\mathrm{ext}} (\tilde{\varphi}_{1}^{i}, { \tilde{\psi}}_{1}^{i}; \frac {x-\tilde {x}^{i}}{r_{\varepsilon, \lambda}} ) + \tilde{v}_{1}(x), \\ u_{\mathrm{ext},2} (x):= \frac{1+ \eta _{3}}{\xi} G(x, \tilde{ x}^{3})+ (1+ \eta _{2}) G(x,\tilde{x}^{2}) + \sum_{i=1}^{3}\chi _{r_{0}} (x- \tilde{ x}^{i}) H^{\mathrm{ext}} (\tilde{\varphi}_{2}^{i}, \tilde{\psi}_{2}^{i}; \frac {x-\tilde {x}^{i}}{r_{\varepsilon, \lambda}} ) + \tilde{v}_{2}(x), \end{cases}\displaystyle \end{aligned}$$
with \(\tilde{v}_{1},\tilde{v}_{2} \in C^{4,\alpha}_{\nu}(\bar{\Omega}^{*}({ \tilde{\mathbf {x}}})) \) satisfying
$$\begin{aligned} \bigl\Vert (\tilde{v}_{1},\tilde{v}_{2}) \bigr\Vert _{(C^{4,\alpha}_{\nu }( \bar{\Omega}^{*}({\tilde{\mathbf {x}}})))^{2}} \leq 2 c_{\kappa} r_{ \varepsilon, \lambda}^{2}. \end{aligned}$$
It remains to determine the parameters and the boundary data in such a way that the function equal to \(u_{\mathrm{int}}\) in \(B_{r_{\varepsilon, \lambda}} (\tilde {x}^{1}) \cup B_{r_{ \varepsilon, \lambda}} (\tilde {x}^{2})\cup B_{r_{\varepsilon }}( \tilde {x}^{3})\) and equal to \(u_{\mathrm{ext}}\) in \(\bar{\Omega}_{{} r_{\varepsilon, \lambda}}(\tilde{\mathbf{x}}) \) is a smooth function. This amounts to find the boundary data and the parameters so that, for each \(j=1, 2\)
$$\begin{aligned} \begin{aligned} &u_{\mathrm{int}, j} = u_{\mathrm{ext}, j},\qquad \partial _{r} u_{\mathrm{int}, j} = \partial _{r} u_{\mathrm{ext}, j},\\ & \Delta u_{\mathrm{int}, j} = \Delta u_{\mathrm{ext}, j} \quad\text{and}\quad \partial _{r} \Delta u_{\mathrm{int}, j} = \partial _{r} \Delta u_{\mathrm{ext}, j} \end{aligned} \end{aligned}$$
(101)
on \(\partial B_{r_{\varepsilon, \lambda}} (\tilde {x}^{1})\), \(\partial B_{r_{\varepsilon, \lambda}}(\tilde {x}^{2})\) and \(\partial B_{r_{\varepsilon, \lambda}}(\tilde {x}^{3})\).
Suppose that (101) is verified, this provides that for each ε small enough \(u_{\varepsilon, \lambda} \in {\mathcal {C}}^{4, \alpha}\) (which is obtained by patching together the functions \(u_{\mathrm{int}}\) and the function \(u_{\mathrm{ext}}\)), a weak solution of our system and elliptic regularity theory implies that this solution is in fact smooth. That will complete the proof, since ε and λ tend to 0, the sequence of solutions we have obtained satisfies the required singular limit behavior.
Before we proceed, the following remarks are due. First it is convenient to observe that the function \(u_{\varepsilon,\tau _{i}}\) can be expanded as
$$\begin{aligned} u_{\varepsilon, \tau _{i}} (x)&= - 4 \ln \tau _{i} - 8 \ln \vert x \vert +{\mathcal {O}} \biggl( \frac{\varepsilon ^{2} \tau _{i}^{-2}}{ \vert x \vert ^{2}} \biggr) \quad\text{on } \partial B_{r_{\varepsilon, \lambda}} ( 0). \end{aligned}$$
(102)
• On \(\partial B_{r_{\varepsilon, \lambda}}( \tilde{x}^{1})\), we have
$$\begin{aligned} &(u_{\mathrm{int},1} - u_{\mathrm{ext},1}) (x) \\ &\quad={ -} \frac{4}{\gamma } \ln \tau _{1} + \frac{8 \eta _{1}}{\gamma } \ln \bigl\vert x-\tilde{x}^{1} \bigr\vert - \frac{1- \gamma }{\gamma \xi } G \bigl(x, \tilde{x}^{3} \bigr) - \frac{\ln \gamma}{\gamma} \\ &\qquad{}+ h_{1}^{1} \biggl(R^{1}_{\varepsilon, \lambda}{ \frac{x- \tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr)+ H^{\mathrm{int}} \biggl(\varphi ^{1}_{1}, \psi _{1}^{1}; { \frac{x-\tilde{x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) - H^{\mathrm{ext}} \biggl( \tilde{\varphi}_{1}^{1}, \tilde{\psi}_{1}^{1}; { \frac{x-\tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr) \\ &\qquad{}- \frac{1 + \eta _{1}}{\gamma}H \bigl( x,\tilde {x}^{1} \bigr) - \biggl(1 + \eta _{2} + \frac{1-\gamma}{\gamma} \biggr) G \bigl( x,\tilde{ x}^{2} \bigr) + \mathcal{O} \biggl( \frac{\varepsilon ^{2} \tau _{1}^{-2}}{ \vert x-\tilde {x}^{1} \vert ^{2}} \biggr) + \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(103)
Next, even though all functions are defined on \(\partial B_{r_{\varepsilon, \lambda}} (\tilde{x}^{1})\) in (101), it will be more convenient to solve on \(S^{3}\) the following set of equations
$$\begin{aligned} \begin{aligned} &(u_{\mathrm{int},1} - u_{\mathrm{ext},1}) \bigl( \tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0,\qquad \partial _{r} (u_{\mathrm{int},1} - u_{\mathrm{ext},1} ) \bigl( \tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0, \\ &\Delta (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) \bigl(\tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) =0\quad \text{and}\quad \partial _{r} \Delta (u_{\mathrm{int},1} - u_{\mathrm{ext},1} ) \bigl(\tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0. \end{aligned} \end{aligned}$$
(104)
Since the boundary data are chosen to satisfy (65) or (66). We decompose
$$\begin{aligned} &\varphi ^{1}_{1} = \varphi ^{1}_{1,0} + \varphi ^{1}_{1,1}+ {\varphi}^{1, \perp}_{1},\qquad \psi ^{1}_{1} = 8 \varphi _{1, 0}^{1} + 12 \varphi _{1, 1}^{1} + \psi _{1}^{1, \perp},\\ & \tilde{\varphi}_{1}^{1} = \tilde{\varphi}^{1}_{1,0} + \tilde{\varphi}^{1}_{1,1}+ \tilde{\varphi}_{1}^{1,\perp} \quad\text{and}\quad \tilde{\psi}_{1}^{1} = \tilde{\psi}_{1, 1}^{1} + \tilde{\psi}_{1}^{1, \perp}, \end{aligned}$$
where \(\varphi ^{1}_{1,0},\tilde{\varphi}^{1}_{1,0 } \in {\mathbb{E}}_{0} = \mathbb{R}\) are constant on \(S^{3}\), \(\varphi ^{1}_{1,1}, \tilde{\varphi}^{1}_{1,1}, \tilde{\psi}_{1, 1}^{1}\) belong to \({\mathbb{E}}_{1} = \operatorname{Span}\{e_{1}, e_{2}, e_{3}, e_{4}\}\) and \({\varphi}^{1,\perp}_{1}, \tilde{\varphi}_{1}^{1,\perp}, {\psi}^{1, \perp}_{1}, \tilde{\psi}_{1}^{1,\perp} \) are \(L^{2} (S^{3})\) orthogonal to \({\mathbb{E}}_{0}\) and \({\mathbb{E}}_{1}\).
Using (103), we have for \(x \in S^{3}\)
$$\begin{aligned} & (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) \bigl(\tilde {x}^{1} + r_{\varepsilon, \lambda} x \bigr) \\ &\quad= - {\frac{4}{\gamma}} \ln \tau _{1} + {\frac{8 \eta _{1}}{\gamma}} \ln \bigl(r_{\varepsilon, \lambda} \vert x \vert \bigr) - \frac{1}{\gamma} \biggl( H \bigl(\tilde{x}^{1}, \tilde{ x}^{1} \bigr)+G \bigl(\tilde{x}^{1}, \tilde{ x}^{2} \bigr) \\ &\qquad{}+ {\frac{1-\gamma}{\xi}}G \bigl(\tilde{x}^{1},\tilde{ x}^{3} \bigr) \biggr)+ H^{\mathrm{int}} \bigl(\varphi ^{1}_{1}, \psi _{1}^{1}; x \bigr) - H^{\mathrm{ext}} \bigl( \tilde{\varphi}_{1}^{1}, \tilde{\psi}_{1}^{1}; x \bigr) \\ &\qquad{}- {\frac{\ln \gamma}{\gamma}}- {\frac{\eta _{1}}{\gamma}}H \bigl( \tilde{x}^{1}, \tilde{ x}^{1} \bigr) -\eta _{2} G \bigl( \tilde{x}^{1}, \tilde{ x}^{2} \bigr) +{\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
Then, the projection of the equations (104) over \({\mathbb{E}}_{0}\) yields
$$\begin{aligned} \textstyle\begin{cases} - 4 \ln \tau _{1} + 8 \eta _{1} \ln r_{\varepsilon }- \ln \gamma + \gamma \varphi _{1,0}^{1} - \gamma \tilde{\varphi}_{1,0}^{1} - \mathcal{E}_{1} (\tilde {x}^{1}, {\tilde{\mathbf {x}}}) + {\mathcal {O}} (r^{2}_{ \varepsilon, \lambda}) = 0, \\ 8 \eta _{1} + 2\gamma \varphi _{1,0}^{1} + 2\gamma \tilde{\varphi}_{1,0}^{1} + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) =0, \\ 16 \eta _{1} + 8 \gamma \varphi _{1,0}^{1} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ -32 \eta _{1} + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) =0, \end{cases}\displaystyle \end{aligned}$$
(105)
where
$$\begin{aligned} \mathcal{E}_{1}( \cdot,{\tilde{\mathbf {x}}}):= H \bigl(\cdot, \tilde{x}^{1} \bigr) + G \bigl( \cdot, \tilde{x}^{2} \bigr)+{ \frac{1-\gamma}{\xi}}G \bigl(\cdot, \tilde{x}^{3} \bigr). \end{aligned}$$
The system (105) can be simply written as
$$\begin{aligned} &\eta _{1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \varphi _{1,0}^{1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\\ & \tilde{\varphi}_{1,0}^{1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr) \quad\text{and}\quad \frac{1}{\ln r_{\varepsilon, \lambda}} \bigl[4 \ln \tau _{1} + \ln \gamma + \mathcal{E}_{1} \bigl(\tilde {x}^{1}, {\tilde{\mathbf {x}}} \bigr) \bigr] = { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
We are now in a position to define \(\tau _{1}^{-}\) and \(\tau _{1}^{+}\). In fact, according to the above analysis, as ε and λ tend to 0, we expect \(\tilde{x}^{i}\) to converge to \(x^{i}\) for \(i \in \{1, 2, 3\}\) and \(\tau _{1}\) to converge to \(\tau _{1}^{*}\), satisfying
$$\begin{aligned} 4 \ln \tau _{1}^{*} = -\ln \gamma - \mathcal{E}_{1} \bigl(x^{1}, {\mathbf{x}} \bigr). \end{aligned}$$
Hence, it is enough to choose \(\tau _{1}^{-}\) and \(\tau _{1}^{+}\) in such a way that
$$\begin{aligned} 4 \ln \bigl(\tau _{1}^{-} \bigr) < - \ln \gamma - \mathcal{E}_{1} \bigl(x^{1}, {\mathbf{x}} \bigr) < 4 \ln \bigl(\tau _{1}^{+} \bigr). \end{aligned}$$
Consider now the projection of (104) over \({\mathbb{E}}_{1}\). Given a smooth function f defined in Ω, we identify its gradient \(\nabla f = (\partial _{x_{1}}f, \ldots,\partial _{x_{4}} f)\) with the element of \({\mathbb{E}}_{1}\)
$$\begin{aligned} \bar{\nabla}f = \sum_{i=1}^{4} \partial _{x_{i}} f e_{i}. \end{aligned}$$
With these notations in mind, we obtain the system of equations
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{1}_{1,1} - \tilde{\varphi}^{1}_{1,1} - \bar{\nabla}\mathcal{E}_{1} (\tilde {x}^{1}, {\tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ 3 \varphi ^{1}_{1,1} + 3 \tilde{\varphi}^{1}_{1,1} + \frac{1}{2} \tilde{\psi}^{1}_{1,1} - \bar{\nabla}\mathcal{E}_{1} (\tilde {x}^{1}, { \tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{1}_{1,1} - 3 \tilde{\varphi}^{1}_{1,1} - \tilde{\psi}^{1}_{1,1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{1}_{1,1} + 15 \tilde{\varphi}^{1}_{1,1} + \frac{18}{4} \tilde{\psi}^{1}_{1,1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \end{cases}\displaystyle \end{aligned}$$
(106)
which can be simplified as follows
$$\begin{aligned} \begin{aligned} &\varphi ^{1}_{1,1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\tilde{\varphi}^{1}_{1,1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\\ & \tilde{\psi}^{1}_{1,1} = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr)\quad \text{and}\quad \bar{\nabla}\mathcal{E}_{1} \bigl(\tilde {x}^{1}, {\tilde{\mathbf {x}}} \bigr) = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned} \end{aligned}$$
(107)
Finally, we consider the projection onto \(L^{2}(S^{3})^{\perp}\). This yields the system
$$\begin{aligned} \textstyle\begin{cases} \varphi _{1}^{1,\perp} - \tilde{\varphi}_{1}^{1,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} (H^{\mathrm{int}}_{\varphi _{1}^{1,\perp}, \psi _{1}^{1, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{1}^{1,\perp}, \tilde{\psi}_{1}^{1, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) = 0, \\ \psi _{1}^{1,\perp} - \tilde{\psi}_{1}^{1,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} \Delta (H^{\mathrm{int}}_{\varphi _{1}^{1,\perp}, \psi _{1}^{1, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{1}^{1,\perp}, \tilde{\psi}_{1}^{1, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(108)
Thanks to the result of Lemma 4, this last system can be rewritten as
$$\begin{aligned} &\varphi _{1}^{1,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\tilde{\varphi}_{1}^{1,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\\ & \psi _{1}^{1,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\quad \text{and}\quad \tilde{\psi}_{1}^{1,\perp}= { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
If we define the parameter \(t_{1} \in \mathbb{R}\) by
$$\begin{aligned} t_{1} = \frac{1}{\ln r_{\varepsilon, \lambda}} \bigl[ 4 \ln \tau _{1} + \ln \gamma + \mathcal{E}_{1} \bigl(\tilde {x}^{1}, \tilde{\mathbf{x}} \bigr) \bigr], \end{aligned}$$
then the systems found by projecting (104) gather in this equality
$$\begin{aligned} T^{1}_{1,\varepsilon } &= \bigl(t_{1}, \eta _{1}, \varphi ^{1}_{1,0}, \tilde{\varphi}^{1}_{1,0}, \varphi _{1,1}^{1}, \tilde{\varphi}_{1,1}^{1}, \tilde{\psi}^{1}_{1,1}, \bar{\nabla}\mathcal{E}_{1} \bigl(\tilde{x}^{1},{ \tilde{\mathbf {x}}} \bigr), \varphi _{1}^{1,\perp},\tilde{\varphi}_{1}^{1,\perp}, \psi _{1}^{1,\perp},\tilde{\psi}_{1}^{1,\perp} \bigr) \\ &= {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr), \end{aligned}$$
(109)
where, as usual, the terms \({\mathcal {O}}(r^{2}_{\varepsilon, \lambda})\) depend nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and κ) times \(r^{2}_{\varepsilon, \lambda}\), provided \(\varepsilon \in (0, \varepsilon _{\kappa})\) and \(\lambda \in (0, \lambda _{\kappa})\).
• On \(\partial B_{r_{\varepsilon }} (\tilde {x}^{1}) \), we have
$$\begin{aligned} (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) (x) ={}&{ -}{ \frac{\eta _{3}}{\xi}} G \bigl(x, \tilde {x}^{3} \bigr) + G \bigl(x, \tilde {x}^{2} \bigr)+ h_{2}^{1} \biggl(R^{1}_{ \varepsilon, \lambda}{ \frac{x-\tilde {x}^{1}}{r_{\varepsilon, \lambda}}} \biggr)+ H^{\mathrm{int}} \biggl(\varphi ^{1}_{2}, \psi ^{1}_{2}; {\frac{x-\tilde{x}^{1}}{r_{ \varepsilon, \lambda}}} \biggr) \\ &{}-(1+\eta _{2})G \bigl(x,\tilde {x}^{2} \bigr)- H^{\mathrm{ext}} \biggl( \tilde{\varphi}^{1}_{2},\tilde{ \psi}^{1}_{2}; {\frac{x-\tilde{x}^{1}}{r_{ \varepsilon, \lambda}}} \biggr)+ \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(110)
In the same manner as above, we will solve on \(S^{3}\) the following system
$$\begin{aligned} \begin{aligned} &(u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl( \tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0,\qquad \partial _{r} (u_{\mathrm{int},2} - u_{\mathrm{ext},2} ) \bigl( \tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0, \\ &\Delta (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl(\tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0 \quad\text{and}\quad \partial _{r} \Delta (u_{\mathrm{int},2} - u_{\mathrm{ext},2} ) \bigl(\tilde{x}^{1} + r_{\varepsilon, \lambda} \cdot \bigr) = 0. \end{aligned} \end{aligned}$$
(111)
We decompose
$$\begin{aligned} &\varphi ^{1}_{2} = \varphi ^{1}_{2,0} + \varphi ^{1}_{2,1}+ {\varphi}^{1, \perp}_{2},\qquad \psi ^{1}_{2} = 8 \varphi _{2, 0}^{1} + 12 \varphi _{2, 1}^{1} + \psi _{2}^{1, \perp},\\ & \tilde{\varphi}_{2}^{1} = \tilde{\varphi}^{1}_{2,0} + \tilde{\varphi}^{1}_{2,1}+ \tilde{\varphi}_{2}^{1,\perp}\quad \text{and} \quad\tilde{\psi}_{2}^{1} = \tilde{\psi}_{2, 1}^{1} + \tilde{\psi}_{2}^{1, \perp}, \end{aligned}$$
with \(\varphi ^{1}_{2,0},\tilde{\varphi}^{1}_{2,0 } \in {\mathbb{E}}_{0}\), \(\varphi ^{1}_{2,1}, \tilde{\varphi}^{1}_{2,1}, \tilde{\psi}_{2, 1}^{1} \in {\mathbb{E}}_{1}\) and \({\varphi}^{1,\perp}_{2}, \tilde{\varphi}_{2}^{1,\perp}, {\psi}^{1, \perp}_{2}, \tilde{\psi}_{2}^{1,\perp} \) belong to \(({L^{2} (S^{3})})^{\bot}\).
Projecting the set of equations (111) over \({\mathbb{E}}_{0}\), we get
$$\begin{aligned} \textstyle\begin{cases} \varphi _{2,0}^{1} - \tilde{\varphi}_{2,0}^{1} + {\mathcal {O}} (r^{2}_{ \varepsilon, \lambda}) = 0, \\ 2\varphi _{2,0}^{1} + 2 \tilde{\varphi}_{2,0}^{1} + {\mathcal {O}} (r^{2}_{ \varepsilon, \lambda}) =0, \\ 8 \varphi _{2,0}^{1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(112)
From the \(L^{2}\)-projection of (111) over \({\mathbb{E}}_{1}\), we obtain the system of equations
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{1}_{2,1} - \tilde{\varphi}^{1}_{2,1}+ {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ 3 \varphi ^{1}_{2,1} + 3 \tilde{\varphi}^{1}_{2,1} + \frac{1}{2} \tilde{\psi}^{1}_{2,1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{1}_{2,1} - 3 \tilde{\varphi}^{1}_{2,1} - \tilde{\psi}^{1}_{2,1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{1}_{2,1} + 15 \tilde{\varphi}^{1}_{2,1} + \frac{18}{4} \tilde{\psi}^{1}_{2,1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(113)
Finally, we consider the \(L^{2}\)-projection onto \((L^{2}(S^{3}))^{\perp}\). This yields the system
$$\begin{aligned} \textstyle\begin{cases} \varphi _{2}^{1,\perp} - \tilde{\varphi}_{2}^{1,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} (H^{\mathrm{int}}_{\varphi _{2}^{1,\perp}, \psi _{2}^{1, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{2}^{1,\perp}, \tilde{\psi}_{2}^{1, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) = 0, \\ \psi _{2}^{1,\perp} - \tilde{\psi}_{2}^{1,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} \Delta (H^{\mathrm{int}}_{\varphi _{2}^{1,\perp}, \psi _{2}^{1, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{2}^{1,\perp}, \tilde{\psi}_{2}^{1, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(114)
Using again Lemma 4, the above system can be rewritten as
$$\begin{aligned} \varphi _{2}^{1,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}_{2}^{1,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\psi _{2}^{1,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr)\quad \text{and}\quad \tilde{\psi}_{2}^{1,\perp}= { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
Then the systems found by projecting (111) gather in this equality
$$\begin{aligned} T^{1}_{2,\varepsilon } = \bigl(\varphi ^{1}_{2,0}, \tilde{\varphi}^{1}_{2,0}, \varphi _{2,1}^{1}, \tilde{\varphi}_{2,1}^{1}, \tilde{\psi}_{2,1}^{1}, \varphi _{2}^{1,\perp},\tilde{\varphi}_{2}^{1,\perp}, \psi _{2}^{1, \perp}, \tilde {\psi}_{2}^{1,\perp} \bigr) = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr). \end{aligned}$$
(115)
• On \(\partial B_{r_{\varepsilon }} (\tilde{x}^{2})\), we have
$$\begin{aligned} &(1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) (x) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) (x) \\ &\quad = - 4(2- \gamma - \xi ) \ln \tau _{2} + 8(2- \gamma - \xi )\eta _{2} \ln \bigl\vert x-\tilde{x}^{2} \bigr\vert \\ &\qquad{}+ (1- \xi ) h_{1}^{2} \biggl(R^{2}_{\varepsilon, \lambda}{ \frac{x- \tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) + (1 - \gamma ) h_{2}^{2} \biggl(R^{2}_{ \varepsilon, \lambda}{\frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \\ &\qquad{}+ (1 - \xi )H^{\mathrm{int}} \biggl(\varphi ^{2}_{1},\psi ^{2}_{1}; {\frac{x- \tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) + (1 - \gamma )H^{\mathrm{int}} \biggl( \varphi ^{2}_{2},\psi ^{2}_{2}; {\frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \\ &\qquad{}- (1- \xi ) H^{\mathrm{ext}} \biggl(\tilde{\varphi}^{2}_{1}, \tilde{\psi}^{2}_{1}; {\frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) -(1 - \gamma )H^{\mathrm{ext}} \biggl( \tilde{\varphi}^{2}_{2}, \tilde{\psi}^{2}_{2}; {\frac{x-\tilde{x}^{2}}{r_{ \varepsilon, \lambda}}} \biggr) \\ &\qquad{}- \biggl[ (2 - \gamma - \xi ) H \bigl( x,\tilde{x}^{2} \bigr) + \frac{1- \xi}{\gamma} G \bigl( x,\tilde{x}^{1} \bigr) + \frac{1 - \gamma}{\xi} G \bigl( x,\tilde{x}^{3} \bigr) \biggr] \\ &\qquad{}+ \mathcal{O} \biggl( \frac{\varepsilon ^{2} \tau _{2}^{-2}}{ \vert x-\tilde {x}^{2} \vert ^{2}} \biggr) + \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(116)
We denote by
$$\begin{aligned} & \varphi ^{2} = (1 - \xi ) \varphi _{1}^{2} + (1 - \gamma ) \varphi _{2}^{2},\qquad \psi ^{2} = (1 - \xi ) \psi _{1}^{2} + (1 - \gamma ) \psi _{2}^{2}, \\ &\tilde {\varphi}^{2} = (1 - \xi ) \tilde{\varphi}^{2}_{1} + (1 - \gamma ) \tilde{\varphi}_{2}^{2}, \qquad\tilde{ \psi}^{2} = (1 - \xi ) \tilde{\psi}^{2}_{1} + (1 - \gamma ) \tilde{\psi}^{2}_{2} \\ &h^{2} = (1 - \xi ) h_{1}^{2} + (1 - \gamma ) h_{2}^{2}, \end{aligned}$$
Then we have
$$\begin{aligned} &(1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) (x) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) (x) \\ &\quad = -4 (2 - \gamma - \xi ) \ln \tau _{2} + 8(2- \gamma - \xi )\eta _{2} \ln \bigl\vert x-\tilde{x}^{2} \bigr\vert + h^{2} \biggl(R^{2}_{\varepsilon, \lambda}{\frac{x- \tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) \\ &\qquad{} + H^{\mathrm{int}} \biggl(\varphi ^{2}, \psi ^{2};{ \frac{x-\tilde{x}^{2}}{r_{\varepsilon, \lambda}}} \biggr) - H^{\mathrm{ext}} \biggl( \tilde{ \varphi}^{2}, \tilde{\psi}^{2}; {\frac{x-\tilde{x}^{2}}{r_{ \varepsilon, \lambda}}} \biggr) \\ &\qquad- \biggl[ (2 - \gamma - \xi ) H \bigl( x,\tilde{x}^{2} \bigr) + \frac{1- \xi}{\gamma} G \bigl( x,\tilde{x}^{1} \bigr) + \frac{1 - \gamma}{\xi} G \bigl( x,\tilde{x}^{3} \bigr) \biggr] \\ &\qquad{}+ \mathcal{O} \biggl( \frac{\varepsilon ^{2} \tau _{2}^{-2}}{ \vert x-\tilde {x}^{2} \vert ^{2}} \biggr)+ \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(117)
Next, even though all functions are defined on \(\partial B_{r_{\varepsilon, \lambda}} (\tilde{x}^{2})\) in (101), it will be more convenient to solve on \(S^{3}\), the following set of equations
$$\begin{aligned} \begin{aligned} &\bigl( (1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigr) \bigl(\tilde{x}^{2} + r_{\varepsilon, \lambda}. \bigr) = 0, \\ &\partial _{r} \bigl( (1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigr) \bigl(\tilde{x}^{2} + r_{\varepsilon, \lambda}. \bigr) = 0, \\ &\Delta \bigl( (1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigr) \bigl(\tilde{x}^{2} + r_{\varepsilon, \lambda}. \bigr) = 0, \\ &\partial _{r} \Delta \bigl( (1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigr) \bigl(\tilde{x}^{2} + r_{ \varepsilon, \lambda}. \bigr) = 0. \end{aligned} \end{aligned}$$
(118)
Since the boundary data are chosen to satisfy (65) or (66). We decompose
$$\begin{aligned} &\varphi ^{2} = \varphi ^{2}_{0} + \varphi ^{2}_{1}+ {\varphi}^{2, \perp}, \qquad\psi ^{2} = 8 \varphi _{0}^{2} + 12 \varphi _{1}^{2} + \psi ^{2, \perp},\\ & \tilde{\varphi}^{2} = \tilde{ \varphi}^{2}_{0} + \tilde{\varphi}^{2}_{1}+ \tilde{\varphi}^{2,\perp} \quad\text{and} \quad\tilde{\psi}^{2} = \tilde{\psi}_{1}^{2} + \tilde{\psi}^{2, \perp}, \end{aligned}$$
where \(\varphi ^{2}_{0},\tilde{\varphi}^{2}_{0 } \in {\mathbb{E}}_{0} = \mathbb{R}\) are constant on \(S^{3}\), \(\varphi ^{2}_{1}, \tilde{\varphi}^{2}_{1}, \tilde{\psi}_{ 1}^{2}\) belong to \({\mathbb{E}}_{1} = \operatorname{Span}\{e_{1}, e_{2}, e_{3}, e_{4}\}\) and \({\varphi}^{2,\perp}, \tilde{\varphi}^{2,\perp}, {\psi}^{2,\perp}, \tilde{\psi}^{2,\perp} \) are \(L^{2} (S^{3})\) orthogonal to \({\mathbb{E}}_{0}\) and \({\mathbb{E}}_{1}\).
We insist that, for \(x \in S^{3}\), both equations (117) involve the same relation of the parameter \(\tau _{2}\) and the appropriate energy \(\mathcal{E}_{2}\). Then we have
$$\begin{aligned} & (1- \xi ) (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) (x) + (1 - \gamma ) (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl( \tilde{x}^{2} + r_{\varepsilon, \lambda}x \bigr) \\ &\quad= - 4 (2 - \gamma - \xi ) \ln \tau _{2} + 8(2- \gamma - \xi )\eta _{2} \ln r_{\varepsilon, \lambda} \vert x \vert + H^{\mathrm{int}} \bigl( \varphi ^{2},\psi ^{2}; x \bigr) - H^{\mathrm{ext}} \bigl( \tilde{\varphi}^{2}, \tilde{\psi}^{2}; x \bigr) \\ &\qquad{}- \biggl[ (2 - \gamma - \xi ) H \bigl( \tilde{x}^{2},\tilde{x}^{2} \bigr) + \frac{1- \xi}{\gamma} G \bigl( \tilde{x}^{2}, \tilde{x}^{1} \bigr) + \frac{1 - \gamma}{\xi} G \bigl(\tilde{x}^{2},\tilde{x}^{3} \bigr) \biggr] + \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(119)
Projecting the set of equations (118) over \({\mathbb{E}}_{0}\), we get
$$\begin{aligned} \textstyle\begin{cases} -4 (2 - \gamma - \xi ) \ln \tau _{2} +8 (2- \gamma - \xi )\eta _{2} \ln r_{\varepsilon, \lambda}+\varphi _{0}^{2} - \tilde{\varphi}_{0}^{2} - \mathcal{E}_{2}(\tilde{x}^{2}, {\tilde{\mathbf {x}}}) +{\mathcal {O}} (r^{2}_{ \varepsilon, \lambda})\\ \quad = 0, \\ 8(2- \gamma - \xi )\eta _{2} +2\varphi _{0}^{2} + 2 \tilde{\varphi}_{0}^{2} + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) =0, \\ 16(2- \gamma - \xi )\eta _{2} + 8 \varphi _{0}^{2} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ -32 (2- \gamma - \xi )\eta _{2} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \end{cases}\displaystyle \end{aligned}$$
(120)
where
$$\begin{aligned} \mathcal{E}_{2}( \cdot,{\tilde{\mathbf {x}}}):= (2-\gamma -\xi ) H \bigl(\cdot, \tilde{x}^{2} \bigr) + \frac{1-\xi}{\gamma}G \bigl(\cdot, \tilde{x}^{1} \bigr)+{\frac{1- \gamma}{\xi}}G \bigl(\cdot, \tilde{x}^{3} \bigr). \end{aligned}$$
The system (120) can be simply written as
$$\begin{aligned} &\eta _{2} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \varphi ^{2}_{0} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \\ &\tilde{\varphi}^{2}_{0} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr)\quad \text{and}\quad \frac{1}{\ln r_{\varepsilon, \lambda}} \biggl[4 \ln \tau _{2} + \frac{\mathcal{E}_{2}(\tilde {x}^{2}, {\tilde{\mathbf {x}}})}{2- \gamma - \xi} \biggr] = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
We are now in a position to define \(\tau _{2}^{-}\) and \(\tau _{2}^{+}\). In fact, according to the above analysis, as ε and λ tend to 0, we expect \(\tilde{x}^{i}\) to converge to \(x^{i}\) for \(i \in \{1, 2, 3\}\) and \(\tau _{2}\) to converge to \(\tau _{2}^{*}\), satisfying
$$\begin{aligned} 4 \ln \tau _{2}^{*} = - \frac{\mathcal{E}_{2}(x^{2}, {\mathbf{x}})}{2-\gamma -\xi}. \end{aligned}$$
Hence it is enough to choose \(\tau _{2}^{-}\) and \(\tau _{2}^{+}\) in such a way that
$$\begin{aligned} 4 \ln \bigl(\tau _{2}^{-} \bigr) < - \frac{\mathcal{E}_{2}(x^{2}, {\mathbf{x}})}{2-\gamma -\xi} < 4 \ln \bigl(\tau _{2}^{+} \bigr). \end{aligned}$$
Consider now the projection of (118) over \({\mathbb{E}}_{1}\). Given a smooth function f defined in Ω, we identify its gradient \(\nabla f = (\partial _{x_{1}}f, \ldots,\partial _{x_{4}} f)\) with the element of \({\mathbb{E}}_{1}\)
$$\begin{aligned} \bar{\nabla}f = \sum_{i=1}^{4} \partial _{x_{i}} f e_{i}. \end{aligned}$$
With these notations in mind, we obtain the system of equations
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{2}_{1} - \tilde{\varphi}^{2}_{1} - \bar{\nabla}\mathcal{E}_{2} (\tilde {x}^{2}, {\tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 3 \varphi ^{2}_{1} + 3 \tilde{\varphi}^{2}_{1} + \frac{1}{2} \tilde{\psi}^{2}_{1} - \bar{\nabla}\mathcal{E}_{2} (\tilde {x}^{2}, { \tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{2}_{1} - 3 \tilde{\varphi}^{2}_{1} -\tilde{\psi}^{2}_{1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{2}_{1} + 15 \tilde{\varphi}^{2}_{1} + \frac{18}{4} \tilde{\psi}^{2}_{1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(121)
Which can be simplified as follows
$$\begin{aligned} \varphi ^{2}_{1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}^{2}_{1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\tilde{\psi}^{2}_{1} = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr)\quad \text{and}\quad \bar{\nabla}\mathcal{E}_{2} \bigl(\tilde {x}^{2}, {\tilde{\mathbf {x}}} \bigr) = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
(122)
Finally, we consider the projection onto \(L^{2}(S^{3})^{\perp}\). This yields the system
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{2,\perp} - \tilde{\varphi}^{2,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} (H^{\mathrm{int}}_{\varphi ^{2,\perp}, \psi ^{2,\perp}} - H^{\mathrm{ext}}_{ \tilde{\varphi}^{2,\perp}, \tilde{\psi}^{2,\perp}} ) + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) = 0, \\ \psi ^{2,\perp} - \tilde{\psi}^{2,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} \Delta (H^{\mathrm{int}}_{\varphi ^{2,\perp}, \psi ^{2,\perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}^{2,\perp}, \tilde{\psi}^{2,\perp}} ) + { \mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(123)
Thanks to the result of Lemma 4, this last system can be rewritten as
$$\begin{aligned} \varphi ^{2,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}^{2,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\psi ^{2,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr) \quad\text{and}\quad \tilde{\psi}^{2,\perp}= {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr). \end{aligned}$$
If we define the parameter \(t_{2} \in \mathbb{R}\) by
$$\begin{aligned} t_{2} = \frac{1}{\ln r_{\varepsilon, \lambda}} \biggl[4 \ln \tau _{2} + \frac{\mathcal{E}_{2}(\tilde {x}^{2}, {\tilde{\mathbf {x}}})}{2- \gamma - \xi} \biggr], \end{aligned}$$
then the systems found by projecting (118) gather in this equality
$$\begin{aligned} T^{2}_{c,\varepsilon } = \bigl(t_{2}, \eta _{2}, \varphi ^{2}_{0}, \tilde{\varphi}^{2}_{0}, \varphi _{1}^{2}, \tilde{\varphi}_{1}^{2}, \tilde{\psi}^{2}_{1}, \bar{\nabla}\mathcal{E}_{2} \bigl(\tilde{x}^{2},{\tilde{\mathbf {x}}} \bigr), \varphi ^{2,\perp}, \tilde{\varphi}^{2,\perp},\psi ^{2, \perp},\tilde{\psi}^{2,\perp} \bigr) = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \end{aligned}$$
(124)
where, as usual, the terms \({\mathcal {O}}(r^{2}_{\varepsilon, \lambda})\) depend nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and κ) times \(r^{2}_{\varepsilon, \lambda}\), provided \(\varepsilon \in (0, \varepsilon _{\kappa})\) and \(\lambda \in (0, \lambda _{\kappa})\).
• On \(\partial B_{r_{\varepsilon }} (\tilde {x}^{3}) \), we have
$$\begin{aligned} &(u_{\mathrm{int},1} - u_{\mathrm{ext},1}) (x) \\ &\quad = {-} \frac{\eta _{1}}{\gamma} G \bigl(x, \tilde {x}^{1} \bigr) - \eta _{2} G \bigl(x,\tilde {x}^{2} \bigr)+ h_{1}^{3} \biggl(R^{3}_{ \varepsilon, \lambda}{\frac{x-\tilde {x}^{3}}{r_{\varepsilon, \lambda}}} \biggr)+ H^{\mathrm{int}} \biggl(\varphi ^{3}_{1},\psi ^{3}_{1}; {\frac{x-\tilde{x}^{3}}{r_{ \varepsilon, \lambda}}} \biggr) \\ &\qquad{} - H^{\mathrm{ext}} \biggl(\tilde{\varphi}^{3}_{1}, \tilde{\psi}^{3}_{1}; {\frac{x-\tilde{x}^{3}}{r_{\varepsilon, \lambda}}} \biggr)+ \mathcal{O} \bigl(r_{ \varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(125)
In the same manner as above, we will solve on \(S^{3}\) the following system
$$\begin{aligned} \begin{aligned}& (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) \bigl( \tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0, \qquad\partial _{r} (u_{\mathrm{int},1} - u_{\mathrm{ext},1} ) \bigl( \tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0, \\ &\Delta (u_{\mathrm{int},1} - u_{\mathrm{ext},1}) \bigl(\tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0 \quad\text{and}\quad \partial _{r} \Delta (u_{\mathrm{int},1} - u_{\mathrm{ext},1} ) \bigl(\tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0. \end{aligned} \end{aligned}$$
(126)
We decompose
$$\begin{aligned} &\varphi ^{3}_{1} = \varphi ^{3}_{1,0} + \varphi ^{3}_{1,1}+ {\varphi}^{3, \perp}_{1},\qquad \psi ^{3}_{1} = 8 \varphi _{1, 0}^{3} + 12 \varphi _{1, 1}^{3} + \psi _{1}^{3, \perp},\\ & \tilde{\varphi}_{1}^{3} = \tilde{\varphi}^{3}_{1,0} + \tilde{\varphi}^{3}_{1,1}+ \tilde{\varphi}_{1}^{3,\perp}\quad \text{and}\quad \tilde{\psi}_{1}^{3} = \tilde{\psi}_{1, 1}^{3} + \tilde{\psi}_{1}^{3, \perp}, \end{aligned}$$
with \(\varphi ^{3}_{1,0},\tilde{\varphi}^{3}_{1,0 } \in {\mathbb{E}}_{0}\), \(\varphi ^{3}_{1,1}, \tilde{\varphi}^{3}_{1,1}, \tilde{\psi}_{1, 1}^{3} \in {\mathbb{E}}_{1}\) and \({\varphi}^{3,\perp}_{1}, \tilde{\varphi}_{1}^{3,\perp}, {\psi}^{3, \perp}_{1}, \tilde{\psi}_{1}^{3,\perp} \) belong to \(({L^{2} (S^{3})})^{\bot}\).
Projecting the set of equations (126) over \({\mathbb{E}}_{0}\), we get
$$\begin{aligned} \textstyle\begin{cases} \varphi _{1,0}^{3} - \tilde{\varphi}_{1,0}^{3} + {\mathcal {O}} (r^{2}_{ \varepsilon, \lambda}) = 0, \\ 2\varphi _{1,0}^{3} + 2 \tilde{\varphi}_{1,0}^{3} + {\mathcal {O}} (r^{2}_{ \varepsilon, \lambda}) =0, \\ 8 \varphi _{1,0}^{3} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(127)
From the \(L^{2}\)-projection of (126) over \({\mathbb{E}}_{1}\), we obtain the system of equations
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{3}_{1,1} - \tilde{\varphi}^{3}_{1,1}+ {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ 3 \varphi ^{3}_{1,1} + 3 \tilde{\varphi}^{3}_{1,1} + \frac{1}{2} \tilde{\psi}^{3}_{1,1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{3}_{1,1} - 3 \tilde{\varphi}^{3}_{1,1} -\tilde{\psi}^{3}_{1,1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{3}_{1,1} + 15 \tilde{\varphi}^{3}_{1,1}+\frac{18}{4} \tilde{\psi}^{3}_{1,1} + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(128)
Finally, we consider the \(L^{2}\)-projection onto \((L^{2}(S^{3}))^{\perp}\). This yields the system
$$\begin{aligned} \textstyle\begin{cases} \varphi _{1}^{3,\perp} - \tilde{\varphi}_{1}^{3,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} (H^{\mathrm{int}}_{\varphi _{1}^{3,\perp}, \psi _{1}^{3, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{1}^{3,\perp}, \tilde{\psi}_{1}^{3, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) = 0, \\ \psi _{1}^{3,\perp} - \tilde{\psi}_{1}^{3,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} \Delta (H^{\mathrm{int}}_{\varphi _{1}^{3,\perp}, \psi _{1}^{3, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{1}^{3,\perp}, \tilde{\psi}_{1}^{3, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(129)
Using again Lemma 4, the above system can be rewritten as
$$\begin{aligned} \varphi _{1}^{3,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}_{1}^{3,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\psi _{1}^{3,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr) \quad\text{and}\quad \tilde{\psi}_{1}^{3,\perp}= { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
Then the systems found by projecting (126) gather in this equality
$$\begin{aligned} T^{3}_{1,\varepsilon } = \bigl(\varphi ^{3}_{1,0}, \tilde{\varphi}^{3}_{1,0}, \varphi _{1,1}^{3}, \tilde{\varphi}_{1,1}^{3}, \tilde{\psi}_{1,1}^{3}, \varphi _{1}^{3,\perp},\tilde{\varphi}_{1}^{3,\perp}, \psi _{1}^{3, \perp}, \tilde{\psi}_{1}^{3,\perp} \bigr) = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr). \end{aligned}$$
(130)
• On \(\partial B_{r_{\varepsilon, \lambda}}( \tilde{x}^{3})\), we have
$$\begin{aligned} &(u_{\mathrm{int},2} - u_{\mathrm{ext},2}) (x) \\ &\quad= - \frac{4}{\xi} \ln \tau _{3} + \frac{8 \eta _{3}}{\xi} \ln \bigl\vert x-\tilde{x}^{3} \bigr\vert - \frac{1- \xi}{\gamma \xi} G \bigl(x, \tilde{x}^{1} \bigr) - \frac{\ln \xi }{\xi } \\ &\qquad{} + h_{2}^{3} \biggl(R^{3}_{\varepsilon, \lambda}{ \frac{x- \tilde {x}^{3}}{r_{\varepsilon, \lambda}}} \biggr)+ H^{\mathrm{int}} \biggl(\varphi ^{3}_{2}, \psi _{2}^{3}; { \frac{x-\tilde{x}^{3}}{r_{\varepsilon, \lambda}}} \biggr) - H^{\mathrm{ext}} \biggl( \tilde{\varphi}_{2}^{3}, \tilde{\psi}_{2}^{3}; { \frac{x-\tilde {x}^{3}}{r_{\varepsilon, \lambda}}} \biggr) \\ &\qquad{} - \frac{1 + \eta _{3}}{\xi}H \bigl( x,\tilde {x}^{3} \bigr) - \biggl(1 + \eta _{2} -\frac{1-\xi}{\xi} \biggr) G \bigl( x,\tilde{ x}^{2} \bigr) + \mathcal{O} \biggl( \frac{\varepsilon ^{2} \tau _{3}^{-2}}{ \vert x-\tilde {x}^{3} \vert ^{2}} \biggr) + \mathcal{O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
(131)
Next, even though all functions are defined on \(\partial B_{r_{\varepsilon, \lambda}} (\tilde{x}^{3})\) in (101), it will be more convenient to solve on \(S^{3}\) the following set of equations
$$\begin{aligned} \begin{aligned}& (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl( \tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0,\qquad \partial _{r} (u_{\mathrm{int},2} - u_{\mathrm{ext},2} ) \bigl( \tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0, \\ &\Delta (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl(\tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) =0 \quad\text{and}\quad \partial _{r} \Delta (u_{\mathrm{int},2} - u_{\mathrm{ext},2} ) \bigl(\tilde{x}^{3} + r_{\varepsilon, \lambda} \cdot \bigr) = 0. \end{aligned} \end{aligned}$$
(132)
Since the boundary data are chosen to satisfy (65) or (66). We decompose
$$\begin{aligned} &\varphi ^{3}_{2} = \varphi ^{3}_{2,0} + \varphi ^{3}_{2,1}+ {\varphi}^{3, \perp}_{2},\qquad \psi ^{3}_{2} = 8 \varphi _{2, 0}^{3} + 12 \varphi _{2, 1}^{3} + \psi _{2}^{3, \perp},\\ & \tilde{\varphi}_{2}^{3} = \tilde{\varphi}^{3}_{2,0} + \tilde{\varphi}^{3}_{2,1}+ \tilde{\varphi}_{2}^{3,\perp} \quad\text{and}\quad \tilde{\psi}_{2}^{3} = \tilde{\psi}_{2, 1}^{3} + \tilde{\psi}_{2}^{3, \perp}, \end{aligned}$$
where \(\varphi ^{3}_{2,0},\tilde{\varphi}^{3}_{2,0 } \in {\mathbb{E}}_{0} = \mathbb{R}\) are constant on \(S^{3}\), \(\varphi ^{3}_{2,1}, \tilde{\varphi}^{3}_{2,1}, \tilde{\psi}_{2, 1}^{3}\) belong to \({\mathbb{E}}_{1} = \operatorname{Span}\{e_{1}, e_{2}, e_{3}, e_{4}\}\) and \({\varphi}^{3,\perp}_{2}, \tilde{\varphi}_{2}^{3,\perp}, {\psi}^{3, \perp}_{2}, \tilde{\psi}_{2}^{3,\perp} \) are \(L^{2} (S^{3})\) orthogonal to \({\mathbb{E}}_{0}\) and \({\mathbb{E}}_{1}\).
Using (131), we have for \(x \in S^{3}\)
$$\begin{aligned} & (u_{\mathrm{int},2} - u_{\mathrm{ext},2}) \bigl(\tilde {x}^{3} + r_{\varepsilon, \lambda} x \bigr) \\ &\quad= - {\frac{4}{\xi}} \ln \tau _{3} + {\frac{8 \eta _{3}}{\xi}} \ln \bigl(r_{\varepsilon, \lambda} \vert x \vert \bigr) - \frac{1}{\xi} \biggl( H \bigl( \tilde{x}^{3}, \tilde{ x}^{3} \bigr)+G \bigl(\tilde{x}^{3},\tilde{ x}^{2} \bigr) \\ &\qquad{}+ {\frac{1-\xi}{\gamma}}G \bigl(\tilde{x}^{3},\tilde{ x}^{1} \bigr) \biggr) + H^{\mathrm{int}} \bigl(\varphi ^{3}_{2}, \psi _{2}^{3}; x \bigr) - H^{\mathrm{ext}} \bigl( \tilde{\varphi}_{2}^{3}, \tilde{\psi}_{2}^{3}; x \bigr) \\ &\qquad{}- {\frac{\ln \xi}{\xi}}- \frac{\eta _{3}}{\xi} H \bigl( \tilde{x}^{3}, \tilde{ x}^{3} \bigr) - \eta _{2} G \bigl( \tilde{x}^{3}, \tilde{ x}^{2} \bigr) +{\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
Then, the projection of the set equations (132) over \({\mathbb{E}}_{0}\) yields
$$\begin{aligned} \textstyle\begin{cases} -4 \ln \tau _{3} + 8 \eta _{3} \ln r_{\varepsilon }- \ln \xi + \xi \varphi _{2,0}^{3} - \xi \tilde{\varphi}_{2,0}^{3} - \mathcal{E}_{3} ( \tilde {x}^{3}, {\tilde{\mathbf {x}}}) + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) = 0, \\ 8 \eta _{3} + 2 \xi \varphi _{2,0}^{3} + 2 \xi \tilde{\varphi}_{2,0}^{3} + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) =0, \\ 16 \eta _{3} + 8 \xi \varphi _{2,0}^{3} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ -32 \eta _{3} + {\mathcal {O}} (r^{2}_{\varepsilon, \lambda}) =0, \end{cases}\displaystyle \end{aligned}$$
(133)
where
$$\begin{aligned} \mathcal{E}_{3}( \cdot,{\tilde{\mathbf {x}}}):= H \bigl(\cdot, \tilde{x}^{3} \bigr) + G \bigl( \cdot, \tilde{x}^{2} \bigr)+{ \frac{1-\xi}{\gamma}}G \bigl(\cdot, \tilde{x}^{1} \bigr). \end{aligned}$$
The system (133) can be simply written as
$$\begin{aligned} &\eta _{3} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\varphi _{2,0}^{3} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \\ &\tilde{\varphi}_{2,0}^{3} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr) \quad\text{and}\quad \frac{1}{\ln r_{\varepsilon, \lambda}} \bigl[4 \ln \tau _{3} + \ln \xi + \mathcal{E}_{3} \bigl(\tilde {x}^{3}, {\tilde{\mathbf {x}}} \bigr) \bigr] = { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
We are now in a position to define \(\tau _{3}^{-}\) and \(\tau _{3}^{+}\). In fact, according to the above analysis, as ε and λ tend to 0, we expect \(\tilde{x}^{i}\) to converge to \(x^{i}\) for \(i \in \{1, 2, 3\}\) and \(\tau _{3}\) to converge to \(\tau _{3}^{*}\), satisfying
$$\begin{aligned} 4 \ln \tau _{3}^{*} = -\ln \xi - \mathcal{E}_{3} \bigl(x^{3}, {\mathbf{x}} \bigr). \end{aligned}$$
Hence it is enough to choose \(\tau _{3}^{-}\) and \(\tau _{3}^{+}\) in such a way that
$$\begin{aligned} 4 \ln \bigl(\tau _{3}^{-} \bigr) < - \ln \xi - \mathcal{E}_{3} \bigl(x^{3}, {\mathbf{x}} \bigr) < 4 \ln \bigl(\tau _{3}^{+} \bigr). \end{aligned}$$
Consider now the projection of (132) over \({\mathbb{E}}_{1}\). Given a smooth function f defined in Ω, we identify its gradient \(\nabla f = (\partial _{x_{1}}f, \ldots,\partial _{x_{4}} f)\) with the element of \({\mathbb{E}}_{1}\)
$$\begin{aligned} \bar{\nabla}f = \sum_{i=1}^{4} \partial _{x_{i}} f e_{i}. \end{aligned}$$
With these notations in mind, we obtain the system of equations
$$\begin{aligned} \textstyle\begin{cases} \varphi ^{3}_{2,1} - \tilde{\varphi}^{3}_{2,1} - \bar{\nabla}\mathcal{E}_{3} (\tilde {x}^{3}, {\tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ 3 \varphi ^{3}_{2,1} + 3 \tilde{\varphi}^{3}_{2,1} + \frac{1}{2} \tilde{\psi}^{3}_{2,1} - \bar{\nabla}\mathcal{E}_{3} (\tilde {x}^{3}, { \tilde{\mathbf {x}}}) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{3}_{2,1} - 3 \tilde{\varphi}^{3}_{2,1} - \tilde{\psi}^{3}_{2,1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \\ 15 \varphi ^{3}_{2,1} + 15 \tilde{\varphi}^{3}_{2,1} + \frac{18}{4} \tilde{\psi}^{3}_{2,1}+ {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0, \end{cases}\displaystyle \end{aligned}$$
(134)
which can be simplified as follows
$$\begin{aligned} \begin{aligned} &\varphi ^{3}_{2,1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}^{3}_{2,1} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \\ &\tilde{\psi}^{3}_{2,1} = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr)\quad \text{and}\quad \bar{\nabla}\mathcal{E}_{3} \bigl(\tilde {x}^{3}, {\tilde{\mathbf {x}}} \bigr) = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned} \end{aligned}$$
(135)
Finally, we consider the projection onto \(L^{2}(S^{3})^{\perp}\). This yields the system
$$\begin{aligned} \textstyle\begin{cases} \varphi _{2}^{3,\perp} - \tilde{\varphi}_{2}^{3,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} (H^{\mathrm{int}}_{\varphi _{2}^{3,\perp}, \psi _{2}^{3, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{2}^{3,\perp}, \tilde{\psi}_{2}^{3, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) = 0, \\ \psi _{2}^{3,\perp} - \tilde{\psi}_{2}^{3,\perp} + {\mathcal {O}}(r^{2}_{ \varepsilon, \lambda}) =0, \\ \partial _{r} \Delta (H^{\mathrm{int}}_{\varphi _{2}^{3,\perp}, \psi _{2}^{3, \perp}} - H^{\mathrm{ext}}_{\tilde{\varphi}_{2}^{3,\perp}, \tilde{\psi}_{2}^{3, \perp}} ) + {\mathcal {O}}(r^{2}_{\varepsilon, \lambda}) =0. \end{cases}\displaystyle \end{aligned}$$
(136)
Thanks to the result of Lemma 4, this last system can be rewritten as
$$\begin{aligned} \varphi _{2}^{3,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr),\qquad \tilde{\varphi}_{2}^{3,\perp}= {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr), \qquad\psi _{2}^{3,\perp} = {\mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr) \quad\text{and}\quad \tilde{\psi}_{2}^{3,\perp}= { \mathcal {O}} \bigl(r^{2}_{\varepsilon, \lambda} \bigr). \end{aligned}$$
If we define the parameter \(t_{3} \in \mathbb{R}\) by
$$\begin{aligned} t_{3} = \frac{1}{ \ln r_{\varepsilon, \lambda}} \bigl[ 4 \ln \tau _{3} + \ln \xi + \mathcal{E}_{3} \bigl(\tilde {x}^{3}, {\tilde{\mathbf {x}}} \bigr) \bigr], \end{aligned}$$
then the systems found by projecting (132) gather in this equality
$$\begin{aligned} T^{3}_{2,\varepsilon } = \bigl(t_{3}, \eta _{3}, \varphi ^{3}_{2,0}, \tilde{\varphi}^{3}_{2,0}, \varphi _{2,1}^{3}, \tilde{\varphi}_{2,1}^{3}, \tilde{\psi}_{2,1}^{3} \bar{\nabla}\mathcal{E}_{3} \bigl(\tilde{x}^{3},{ \tilde{\mathbf {x}}} \bigr), \varphi _{2}^{3,\perp},\tilde{\varphi}_{2}^{3,\perp}, \psi _{2}^{3,\perp},\tilde{\psi}_{2}^{3,\perp} \bigr) = {\mathcal {O}} \bigl(r^{2}_{ \varepsilon, \lambda} \bigr), \end{aligned}$$
(137)
where, as usual, the terms \({\mathcal {O}}(r^{2}_{\varepsilon, \lambda})\) depend nonlinearly on all the variables on the left side, but are bounded (in the appropriate norm) by a constant (independent of ε and κ) times \(r^{2}_{\varepsilon, \lambda}\), provided \(\varepsilon \in (0, \varepsilon _{\kappa})\) and \(\lambda \in (0, \lambda _{\kappa})\).
We recall that \({\mathbf{d}} = r_{\varepsilon, \lambda} (\tilde {\mathbf{x}} - {\mathbf{x}})\), in addition the previous systems can be written as for \(i=1,2,3\):
$$\begin{aligned} \bigl({\mathbf{d}}, t_{i}, \eta _{i}, \varphi ^{i}, \tilde{\varphi}^{i}, \psi ^{i}, \tilde{\psi}^{i}, \bar{\nabla}{\mathcal {E}_{i}} \bigr) = {\mathcal {O}} \bigl(r_{\varepsilon, \lambda}^{2} \bigr). \end{aligned}$$
Combining (109), (115), (124), (130), and (137), we have
$$\begin{aligned} T_{i,\varepsilon } = \bigl( T^{1}_{i,\varepsilon },T^{2}_{i,\varepsilon },T^{3}_{i, \varepsilon } \bigr) = \bigl( \mathcal{ O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr), \mathcal{ O} \bigl(r_{\varepsilon, \lambda}^{2} \bigr), \mathcal{ O} \bigl(r_{ \varepsilon, \lambda}^{2} \bigr) \bigr), \quad\text{for } i=1,2. \end{aligned}$$
(138)
Then the nonlinear mapping, which appears on the right-hand side of (138), is continuous and compact. In addition, reducing \(\varepsilon _{\kappa}\) and \(\lambda _{\kappa}\), if necessary, this nonlinear mapping sends the ball of radius \(\kappa r^{2}_{\varepsilon, \lambda}\) (for the natural product norm) into itself, provided κ is fixed large enough. Applying Schauder’s fixed point theorem in the ball of radius \(\kappa r_{\varepsilon, \lambda}^{2}\) in the product space where the entries live, we obtain the existence of a solution of equation (138).
This completes the proof of Theorem 5. □