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Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms
Boundary Value Problems volume 2019, Article number: 131 (2019)
Abstract
Given a bounded open regular set \(\varOmega \in \mathbb{R}^{4}, x_{1}, x_{2}, \ldots, x_{m} \in \varOmega, \lambda, \rho > 0, \gamma \in (0,1)\), and \({\mathscr{Q}}_{\lambda }\) some nonlinear operator (which will be defined later), we prove that the problem
has a positive weak solution in Ω with \(u = \Delta u=0\) on ∂Ω, which is singular at each \(x_{i}\) as the parameters λ and ρ tend to 0.
1 Introduction and statement of results
Semilinear equations involving fourth order elliptic operator and exponential nonlinearity appear naturally in conformal geometry and in particular in the prescription of the so-called Q-curvature in four-dimensional Riemannian manifolds [9, 10]
where \(\operatorname{Ric}_{g}\) denotes the Ricci tensor and \(S_{g}\) is the scalar curvature of the metric g. Recall that the Q-curvature changes under a conformal change of metric
according to
where
is the Paneitz operator, which is an elliptic fourth order partial differential operator [10] and which transforms according to
under a conformal change of metric \(g_{w}: = e^{2w} g\). In the special case where the manifold is the Euclidean space, the Paneitz operator is simply given by
in which case (1) can be written as
the solutions of which give rise to the conformal metric \(g_{w} = e ^{2 w} g_{\mathrm{eucl}}\) whose Q-curvature is given by Q̃. There is by now extensive literature about this problem, and we refer to [10] and [15] for references and recent developments.
In this paper, we are interested in positive solutions of
with a regular bounded open domain \(\varOmega \subset {\mathbb{R}}^{4}\), \(\lambda, \rho > 0, \gamma \in (0,1) \) and a nonlinear operator \({\mathscr{Q}}_{\lambda }\) given by
Using the following transformation:
so, if u is a solution of (2), then w solves the following equation:
where \(\theta = (\lambda \rho ^{4})^{1 - \gamma }\). Remark that the exponent \({q = \frac{\lambda + 1}{\lambda }}\) tends to ∞ as λ tends to 0.
We denote by ε the smallest positive parameter satisfying
Remark that \(\rho \sim \varepsilon \) as \(\varepsilon \longrightarrow 0\). We will suppose in the following that
In particular, if we take \(\lambda = \mathcal{O}(\varepsilon ^{2/3})\), then condition \((A_{\lambda })\) is satisfied. Under assumption \((A_{\lambda })\), we can treat equation (2) as a perturbation of the following equation:
Our question is as follows: Does there exist \(u_{\varepsilon }\) a sequence of solutions which converges to some singular function as the parameters ε and λ tend to 0?
In [2], Baraket et al. gave a positive answer to the above question for the following problem:
with a regular bounded open domain \(\varOmega \subset {\mathbb{R}}^{4}\). They gave a sufficient condition for problem (7) to have a weak solution in Ω which is singular at some points \((x_{i})_{1 \leq i\leq m}\) as ρ and λ are small parameters satisfying \((A_{\lambda })\).
In case \(\lambda = 0 \), the authors in [3] gave also a positive answer to the following problem:
for \(\gamma \in (0,1)\) as ρ tends to 0. When \(\lambda = 0 \) and \(\gamma = 0 \), problem (2) reduces to
In 1996, Wei in [20] studied the behavior of solutions to the following nonlinear problem in \(\mathbb{R}^{4}\). More precisely, consider the following problem:
Before showing his result, we will introduce some notations. Let \(G(x,x')\) defined over \(\varOmega \times \varOmega \), the Green’s function associated with the bi-Laplacian operator with a Navier boundary conditions, which is the solution of
and denote by \(H(x,x')=G(x,x')+8\log |x-x'|\) its smooth part. Consider now the functional
Denote by \(u^{*}\) the solution of
The author proved the following result.
Theorem 1
([20])
Let Ω be a smooth bounded domain in \(\mathbb{R} ^{4}\) and f be a smooth nonnegative increasing function such that
For \(u_{\lambda }\) solution of (10), denote by \({\varSigma _{\lambda } = \lambda \int _{\varOmega }f(u_{\lambda }) \,dx}\). Then three cases occur:
-
1.
\(\varSigma _{\lambda } \longrightarrow 0\), therefore \(\|u_{ \lambda }\|_{L^{\infty (\varOmega )}} \longrightarrow 0\) as \(\lambda \longrightarrow 0\).
-
2.
\(\varSigma _{\lambda } \longrightarrow +\infty \), then \(u_{\lambda }\longrightarrow +\infty \) as \(\lambda \longrightarrow 0\).
-
3.
\(\varSigma _{\lambda } \longrightarrow 64 \pi ^{2} m\) for some positive integer m. Then the limiting function \(u^{*} = { \lim_{\lambda \longrightarrow 0} u_{\lambda }}\) has m blow-up points \(\{x^{1},\ldots, x^{m}\}\), where \(u_{\lambda }(x^{i})\longrightarrow + \infty \) as \(\lambda \longrightarrow 0\). Moreover, \((x^{1},\ldots, x ^{m})\) is a critical point of E.
Recently, in [4], the authors constructed non-minimal solutions of problem (9) with singular limit as the parameter ρ tends to 0. Their results, which give the converse of case \((3)\) given in the last theorem, can be stated as follows.
Theorem 2
([4])
Let Ω be a smooth open subset of \(\mathbb{R} ^{4}\). Assume that \((x^{1},\ldots,x^{m})\) is a nondegenerate critical point of E, then there exist \(\rho _{0} > 0\) and a one-parameter family \((u_{\rho })_{\rho \in (0, \rho _{0}) }\) of solutions of (9) such that
Our main result is the following.
Theorem 3
Given \(\alpha \in (0,1)\). Let Ω be an open smooth bounded set of \(\mathbb{{\mathbb{R}}}^{4}\), \(\lambda >0\) satisfying condition \((A_{\lambda })\), and \(S=\{x_{1},\ldots,x_{m}\}\subset \varOmega \) be a nonempty set. Assume that \((x_{1},\ldots,x_{m})\) is a nondegenerate critical point of the function
then there exist \(\rho _{0} > 0, \lambda _{0} >0\), and a family \({ \{u_{\rho, \lambda } \}}_{ 0 < \rho < \rho _{0}, 0 < \lambda < \lambda _{0} } \) of solutions of (2) such that
2 Construction of the approximate solution
We first describe the rotationally symmetric approximate solutions of
in \(\mathbb{R}^{4}\), which will be crucial in the construction of the approximate solution. Given \(\varepsilon >0\), we define
which is clearly a solution of (14) when
For \(\tau > 0\), we remark that equation (14) is invariant under some dilation in the following sense: If u is a solution of (14), then
is also solution of (14). So, for \(\varepsilon >0\) and \(\tau > 0\), we denote by \(u_{\varepsilon,\tau }\) the element of this new family of radial solutions of (14).
For \(\varepsilon =\tau =1\), we denote by \(u_{1} = u_{1,1}\) this particular solution. We also define the following linear fourth order elliptic operator:
which corresponds to the linearization of (14) about the solution \(u_{1}\).
2.1 Radial solution on \(\mathbb{R}^{4}\)
For all \(\varepsilon, \tau, \lambda > 0, \delta \in (0,1)\), we set
where
The classification of bounded solutions of \({\mathbb{L}} w =0\) in \({\mathbb{R}}^{4}\) is well known. Some solutions are easy to find. For example, we can define
where \(r=|x|\). Clearly, \({\mathbb{L}} \phi _{0} =0\), and this reflects the fact that (14) is invariant under the group of dilations \(\tau \mapsto u(\tau \cdot ) + 4 \log \tau \). We also define, for \(i=1, \ldots, 4\),
which are also solutions of \({\mathbb{L}} \phi _{i} =0\) since these solutions correspond to the invariance of the equation under the group of translations \(a \mapsto u( \cdot +a)\).
Then we have the following classification.
Lemma 1
([4])
Any bounded solution of \({ \mathscr{L}} w = 0\) defined in \({\mathbb{R}}^{4}\) is a linear combination of \(\phi _{i}\) for \(i=0, 1, \ldots,4\).
Let \(B_{r}\) denote the ball of radius r centered at the origin in \({\mathbb{R}}^{4}\).
Definition 1
Given \(k \in {\mathbb{N}}\), \(\alpha \in (0,1)\) and \(\mu \in {\mathbb{R}}\), we introduce the Hölder weighted spaces \({\mathcal{C}}^{k,\alpha }_{\mu }(\mathbb{R}^{4})\) as the space of functions \(w \in {\mathcal{C}}^{k,\alpha }_{\mathrm{loc}}(\mathbb{R} ^{4})\) for which the following norm
is finite.
We also define
As a consequence of the result of Lemma 1, we recall the surjectivity result of \({\mathscr{L}}\).
Proposition 1
([4])
-
1.
Assume that \(\mu > 1\) and \(\mu \notin \mathbb{Z}\), then
$$\begin{aligned} &L_{\mu }:\quad {\mathcal{C}}^{4,\alpha }_{\mu } \bigl(\mathbb{R}^{4}\bigr) \longrightarrow {\mathcal{C}}^{0,\alpha }_{\mu -4} \bigl(\mathbb{R}^{4}\bigr), \\ &\phantom{L_{\mu }:\quad}w \longmapsto { \mathscr{L}} w \end{aligned}$$is surjective.
-
2.
Assume that \(\delta > 0\) and \(\delta \notin \mathbb{Z}\), then
$$\begin{aligned} &L_{\delta }:\quad \mathcal{C}^{4,\alpha }_{\mathrm{rad},\delta } \bigl( \mathbb{R}^{4}\bigr)) \longrightarrow \mathcal{C}^{0,\alpha }_{ \mathrm{rad},\delta -4} \bigl(\mathbb{R}^{4}\bigr), \\ &\phantom{L_{\delta }:\quad}w \longmapsto { \mathscr{L}} w \end{aligned}$$is surjective.
We set \(\bar{B}_{1}^{*} = \bar{B}_{1} - \{0\}\).
Definition 2
Given \(k\in {\mathbb{N}}\), \(\alpha \in (0,1)\), and \(\mu \in {\mathbb{R}}\), we introduce the Hölder weighted spaces \({\mathcal{C}}^{k,\alpha }_{\mu }(\bar{B}_{1}^{*})\) as the space of functions in \({\mathcal{C}}^{k,\alpha }_{\mathrm{loc}}(\bar{B}_{1} ^{*})\) for which the following norm
is finite.
Then we define the subspace of radial functions in \(\mathcal{C}^{k, \alpha }_{\delta }(\bar{B}_{1}^{*})\) by
Our aim now is the construction of a radial solution u of
Thanks to the following transformation
then Eq. (16) can be written as
Now, we look for a solution of (17) of the form
this amounts to solving
in \(\bar{B}_{R_{\varepsilon,\lambda }}\). We will need the following definition.
Definition 3
Given \(\bar{r} \geqslant 1\), \(k \in {\mathbb{N}}\), \(\alpha \in (0,1)\), and \(\delta \in {\mathbb{R}}\), the weighted space \({\mathcal{C}}^{k, \alpha }_{\delta } (B_{\bar{r}})\) is defined to be the space of functions \(w \in {{\mathcal{C}}}^{k, \alpha } (B_{\bar{r}})\) endowed with the norm
For all \(\sigma \geqslant 1\), we denote by
the extension operator defined by
where \(t \longmapsto \chi (t)\) is a smooth nonnegative cut-off function identically equal to 1 for \(t \geqslant 2\) and identically equal to 0 for \(t \leqslant 1\). It is easy to check that there exists a constant \(c = c (\delta ) >0\), independent of \(\sigma \geqslant 1\), such that
We fix
and denote by \({\mathscr{G}}_{\delta }\) a right inverse of \(\mathscr{L}_{\delta }\) assured by Proposition 1. Now, we use the result of Proposition 1 to rephrase the nonlinear equation (18) as a fixed point problem. Hence, to obtain a solution of (18), it is enough to find a fixed point h in a small ball of \(\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }( \mathbb{R}^{4})\) for the mapping
where
We have
Recall that
then
Hence,
and
This implies that given \(\kappa > 0\), there exists \(c_{\kappa }>0\) (which can depend only on κ) such that, for \(\delta \in (0,1)\) and \(|x| = r\), we have
So
Making use of Proposition 1 together with (19), we deduce that
Now let \(h_{1}, h_{2} \text{ in } B (0, 2 c_{\kappa } r_{\varepsilon ,\lambda }^{2})\) of \({\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }(\mathbb{R}^{4})}\) and for \(\delta \in (0,1)\), then
Furthermore,
-
$$\begin{aligned} &\sup_{r \leqslant R_{\varepsilon,\lambda }} r^{4-\delta } \bigl(1 + \vert x \vert ^{2}\bigr)^{-4 } { \bigl\vert e^{h_{2}}-e^{h_{1}}+ h_{1}-h_{2} \bigr\vert }\\ &\quad \leqslant c \sup_{r \leqslant R_{\varepsilon,\lambda }} r^{4-\delta } \bigl(1 + \vert x \vert ^{2}\bigr)^{-4 } \vert h_{2}-h_{1} \vert \vert h_{2}+h_{1} \vert \\ &\quad\leqslant c_{\kappa } r^{\delta } r^{2}_{\varepsilon,\lambda } \Vert h_{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }( \mathbb{R}^{4})}. \end{aligned}$$
-
$$\begin{aligned} &\sup_{r \leqslant R_{\varepsilon,\lambda }} r^{4-\delta } \varepsilon ^{8(1 - \gamma )} \bigl(1 + \vert x \vert ^{2} \bigr)^{-4 } { \bigl\vert e^{ \gamma h_{2}}-e^{ \gamma h_{1}} \bigr\vert }\\ &\quad\leqslant c \sup_{r \leqslant R_{\varepsilon,\lambda }} r^{4-\delta } \varepsilon ^{8(1 - \gamma )}\bigl(1 + \vert x \vert ^{2} \bigr)^{-4 } \vert h_{2}-h_{1} \vert \\ &\quad\leqslant c_{\kappa } \max \bigl\{ \varepsilon ^{8(1 - \gamma )}, \varepsilon ^{4 }r^{-4 }_{\varepsilon,\lambda }\bigr\} \Vert h_{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }(\mathbb{R}^{4})}. \end{aligned}$$
-
$$\begin{aligned} &r^{4-\delta } { \bigl\vert \bigl(\Delta (u_{1}+h_{1}) \bigr)^{2} -\bigl(\Delta ( u_{1}+h _{2}) \bigr)^{2} \bigr\vert }\\ &\quad= r^{4-\delta } { \bigl\vert \bigl( \Delta (h_{1}-h_{2})\bigr) \bigl(\Delta (2u_{1}+h_{1}+h_{2})\bigr) \bigr\vert } \\ &\quad\leqslant c_{\kappa } { \bigl( 1+r^{\delta } r^{2}_{\varepsilon ,\lambda } \bigr)} \Vert h_{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{ \mathrm{rad}, \delta }(\mathbb{R}^{4})}. \end{aligned}$$
-
$$\begin{aligned} &r^{4-\delta } { \bigl\vert \Delta \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}- \Delta \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2} \bigr\vert }\\ &\quad= r^{4-\delta } { \bigl\vert \Delta \bigl(\nabla (h_{1}-h_{2}).\nabla (2u_{1}+h_{1}+h_{2}) \bigr) \bigr\vert } \\ &\quad\leqslant c_{\kappa } { \bigl( 1+r^{\delta } r^{2}_{\varepsilon ,\lambda } \bigr)} \Vert h_{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{ \mathrm{rad}, \delta }(\mathbb{R}^{4})}. \end{aligned}$$
-
\(r^{4-\delta } \vert \nabla (\Delta (u_{1}+h_{2})).\nabla ( u_{1}+h _{2})- \nabla (\Delta (u_{1}+h_{1})).\nabla ( u_{1}+h_{1}) \vert = r^{4-\delta } \vert \nabla (\Delta (h_{1}-h_{2})).\nabla (2u _{1}+h_{1}+h_{2})+ \nabla (h_{2}-h_{1}).\nabla (\Delta (2u_{1}+h_{1}+h _{2})) \vert r^{4-\delta } \vert \nabla (\Delta (u_{1}+h_{2})).\nabla ( u _{1}+h_{2})- \nabla (\Delta (u_{1}+h_{1})).\nabla ( u_{1}+h_{1}) \vert \leqslant c_{\kappa } ( 1+r^{\delta } r^{2}_{\varepsilon, \lambda } ) \|h_{2}-h_{1}\|_{\mathcal{C}^{4,\alpha }_{ \mathrm{rad}, \delta }(\mathbb{R}^{4})}\).
-
Since
$$\begin{aligned} & \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}\Delta (u_{1} + h_{1})- \bigl\vert \nabla (u_{1}+h _{2}) \bigr\vert ^{2}\Delta (u_{1}+h_{2}) \\ &\quad = \Delta (h_{1}-h_{2}) \bigl[ \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}+ \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2} \bigr] \\ &\qquad{}+ \Delta (2 u_{1}+h_{1}+h_{2}) \bigl[ \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}- \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2} \bigr], \end{aligned}$$then
$$\begin{aligned} &r^{4-\delta } \bigl\vert \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}\Delta (u_{1}+h_{1})- \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}\Delta (u_{1}+h_{2}) \bigr\vert \\ &\quad \leqslant c_{\kappa } { \bigl(1+r^{\delta } r^{2}_{\varepsilon ,\lambda }+r^{2\delta } r_{\varepsilon,\lambda }^{4} \bigr)} \Vert h _{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }( \mathbb{R}^{4})}. \end{aligned}$$ -
It is easy to see that
$$\begin{aligned} &\nabla \bigl( \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}\bigr)\nabla (u_{1}+h_{2})- \nabla \bigl( \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}\bigr)\nabla (u_{1}+h_{1}) \\ & \quad= \nabla (h_{2}-h_{1})\nabla \bigl( \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}+ \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2} \bigr) \\ &\qquad{}+ \nabla (2 u_{1}+h_{1}+h_{2})\nabla \bigl( \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}- \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2} \bigr). \end{aligned}$$Hence,
$$\begin{aligned} &r^{4-\delta } \bigl\vert \nabla \bigl( \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}\bigr)\nabla (u_{1}+h _{2})- \nabla \bigl( \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2}\bigr)\nabla (u_{1}+h_{1}) \bigr\vert \\ &\quad \leqslant c_{\kappa } { \bigl(1+r^{\delta } r^{2}_{\varepsilon ,\lambda }+r^{2\delta } r_{\varepsilon,\lambda }^{4} \bigr)} \Vert h _{2}-h_{1} \Vert _{\mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }( \mathbb{R}^{4})}. \end{aligned}$$ -
Finally, since
$$\begin{aligned} & \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{4}- \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{4} \\ &\quad = \nabla (h_{2}-h_{1})\nabla (2u_{1}+h_{2}+h_{1}) \bigl( \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{2}+ \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{2} \bigr), \end{aligned}$$then
$$\begin{aligned} &r^{4-\delta } \bigl\vert \bigl\vert \nabla (u_{1}+h_{2}) \bigr\vert ^{4}- \bigl\vert \nabla (u_{1}+h_{1}) \bigr\vert ^{4} \bigr\vert \\ &\quad \leqslant c_{\kappa } { \bigl(1+r^{\delta } r^{2}_{\varepsilon ,\lambda }+r^{2\delta } r_{\varepsilon,\lambda }^{4}+r^{3\delta } r _{\varepsilon,\lambda }^{6} \bigr)} \Vert h_{2}-h_{1} \Vert _{\mathcal{C} ^{4,\alpha }_{\mathrm{rad}, \delta }(\mathbb{R}^{4})}. \end{aligned}$$
Besides, thanks to condition \((A_{\lambda })\) we deduce that
Similarly, making use of Proposition 1 together with (19), we conclude that, given \(\kappa >0\), there exist \(\bar{c}_{\kappa } >0\) (independent of ε and λ), \(\lambda _{\kappa }\), and \(\varepsilon _{\kappa }\) such that
Reducing \(\lambda _{\kappa }>0\) and \(\varepsilon _{\kappa } > 0\) if necessary, we can assume that \(\bar{c}_{\kappa } r^{2}_{\varepsilon ,\lambda } \leqslant \frac{1}{2} \nonumber \) for all \(\lambda \in (0, \lambda _{\kappa })\) and \(\varepsilon \in (0, \varepsilon _{\kappa })\). Then (24) and (23) are enough to show that \(h \longmapsto {\mathscr{N}} (h)\) is a contraction from \(\{ h \in {\mathcal{C}}^{4, \alpha }_{\mathrm{rad}, \delta }({\mathbb{R}}^{4}): \|h \|_{{\mathcal{C}}^{4, \alpha } _{\mathrm{rad}, \delta }({\mathbb{R}}^{4})} \leqslant 2 c_{\kappa } r^{2}_{\varepsilon,\lambda } \} \) into itself and hence has a unique fixed point h in this set. This fixed point is a solution of (20) in \(\bar{B}_{R_{\varepsilon,\lambda }}\). We summarize this in the following proposition.
Proposition 2
Given \(\delta \in (0,1)\) and \(\kappa >0\), there exist \(\varepsilon _{\kappa }> 0, \lambda _{\kappa }> 0\), and \(\bar{c}_{ \kappa } >0\) (depending on κ) such that, for all \(\lambda \in (0, \lambda _{\kappa })\) and for \(\varepsilon \in (0, \varepsilon _{\kappa })\), there exists a unique solution \(h \in \mathcal{C}^{4,\alpha }_{\mathrm{rad}, \delta }(\mathbb{R}^{4})\) of (20) such that
solves (17) in \(\bar{B}_{R_{\varepsilon,\lambda }}\). In addition,
2.2 Analysis of the bi-Laplace operator in weighted spaces
In this section, we prove a surjectivity result of the bi-Laplace operator in some weighted spaces and recall some estimations concerning the bi-harmonic extensions.
First, given \(x^{1}, \ldots, x^{m} \in \varOmega \), we define
and we choose \(r_{0} >0\) so that the balls \(B_{r_{0}}(x^{i})\) of center \(x^{i}\) and radius \(r_{0}\) are mutually disjoint and included in Ω. Given \(k \in \mathbb{N}\), \(\alpha \in (0,1)\), and \(\nu \in \mathbb{R}\), we introduce the Hölder weighted space \({\mathcal{C}}^{k,\alpha }_{\nu }(\bar{\varOmega }^{*})\) as the space of functions \(w \in {\mathcal{C}}^{k,\alpha }_{\mathrm{loc}}(\bar{\varOmega }^{*})\) which is endowed with the norm
When \(k \geqslant 2\), we let \([ {\mathcal{C}}^{k,\alpha }_{\nu }(\bar{ \varOmega }^{*}) ]_{0}\) be the subspace of functions \(w \in {\mathcal{C}} ^{k,\alpha }_{\nu }(\bar{\varOmega }^{*})\) satisfying \(w = \Delta w = 0\).
In the rest of the paper, we need the following mapping properties of \(\Delta ^{2}\).
Proposition 3
([4])
Assume that \(\nu < 0\) and \(\nu \notin \mathbb{Z}\), then
is surjective.
Remark 1
([4])
It will be interesting to observe that when \(\nu <0\), \(\nu \notin {\mathbb{Z}}\), the right inverse, even though it is not unique, can be chosen to depend smoothly on the points \(x^{1}, \ldots , x^{m}\), at least locally. Once a right inverse is fixed for some choice of the points \(x^{1}, \ldots, x^{m}\), a right inverse which depends smoothly on the points \(\tilde{x}^{1}, \ldots, \tilde{x}^{m}\) close to \(x^{1}, \ldots, x^{m}\) can be obtained using a simple perturbation argument.
Proof
Indeed, given \((\tilde{x}^{i})\) close enough to \((x^{i})\), we can define a family of diffeomorphisms \(D: \varOmega \rightarrow \varOmega \) depending smoothly on \((\tilde{x}^{i})\) by
where \(\chi _{r_{0}}\) is a cut-off function identically equal to 1 in \(B_{r_{0}/2}\) and identically equal to 0 outside \(B_{r_{0}}\). Hence \(D (\tilde{x}^{j}) = x^{j}\) for each j. Then the equation \(\Delta ^{2} \tilde{w} = \tilde{f}\) where \(\tilde{f} \in {\mathcal{C}} ^{0, \alpha }_{\nu -4} (\bar{\varOmega }- \{\tilde{x}^{i}, 1 \leqslant i \leqslant m\})\) can be solved by considering \(\tilde{w} = w \circ D\), where w is a solution of the problem
and this time \(\tilde{f} \circ D^{-1} \in {\mathcal{C}}^{0, \alpha } _{\nu }(\bar{\varOmega }- \{x^{1}, \ldots, x^{m}\})\). It should be clear that
Since we have a fixed right inverse for \(\Delta ^{2}:{\mathcal{C}} ^{4, \alpha }_{\nu }(\bar{\varOmega }^{*}) \rightarrow {\mathcal{C}}^{0, \alpha }_{\nu -4} (\bar{\varOmega }^{*})\), a perturbation argument shows that (25) is solvable provided the \(\tilde{x}^{j}\) are close enough to the \(x^{j}\). This provides a right inverse which depends smoothly on the choice of the points \(\tilde{x}^{i}\). □
2.3 Bi-harmonic extensions
Now, we give some estimates about the bi-harmonic extensions. More precisely, given \(\varphi \in {\mathcal{C}}^{4,\alpha }(S^{3})\) and \(\psi \in {\mathcal{C}}^{2,\alpha }(S^{3})\), we define \(H^{i} (= H ^{i}_{\varphi, \psi })\) to be the solution of
where, as already mentioned, \(B_{1}\) denotes the unit ball in \({\mathbb{R}}^{4}\). Given \(k\in {\mathbb{N}}\), \(\alpha \in (0,1)\), and \(\nu \in {\mathbb{R}}\), we introduce the Hölder weighted spaces \({\mathcal{C}}^{k,\alpha }_{\nu }(\bar{B}_{1}^{*})\) as the space of function in \({\mathcal{C}}^{k,\alpha }_{\mathrm{loc}}(\bar{B}_{1}^{*})\) for which the following norm
is finite. Here \(\bar{B}_{1}^{*} = \bar{B}_{1} - \{0\}\), therefore this norm corresponds to the norm already defined in the previous section when \(\varOmega = B_{1}\), \(m=1\), and \(x^{1}=0\). We denote by \(e_{1}, \ldots, e_{4}\) the coordinate functions on \(S^{3}\).
Lemma 2
([4])
Assume that
for \(\ell = 1, \ldots, 4\). Then there exists \(c> 0\) such that
Also, given \(\varphi \in {\mathcal{C}}^{4,\alpha }(S^{3})\) and \(\psi \in {\mathcal{C}}^{2,\alpha }(S^{3})\), we define (when it exists !) \(H^{e}( =H^{e}_{\varphi, \psi })\) to be a solution of
which decays at infinity. Given \(k\in {\mathbb{N}}\), \(\alpha \in (0,1)\), and \(\mu \in {\mathbb{R}}\), we introduce the Hölder weighted spaces \({\mathcal{C}}^{k,\alpha }_{\mu }({\mathbb{R}}^{4} -B_{1})\) as the space of function \(w \in {\mathcal{C}}^{k,\alpha }_{\mathrm{loc}}({\mathbb{R}} ^{4} -B_{1})\) for which the following norm
is finite.
Lemma 3
([4])
Assume that
Then there exists \(c > 0\) such that
Observe that, under the hypothesis of the lemma, there is uniqueness of the bi-harmonic extension of the boundary data which decays at infinity.
If \(E\subset L^{2}(S^{3})\) is a space of functions defined on \(S^{3}\), we define the space \(E^{\perp }\) to be the subspace of functions which are \(L^{2}\)-orthogonal to the functions \(1, e_{1}, \ldots, e_{4}\).
Lemma 4
([4])
The mapping
is an isomorphism.
3 The nonlinear interior problem
We are interested in studying equations of type
in \(\bar{B}_{R_{\varepsilon,\lambda }}\).
Given \(\varphi \in {\mathcal{C}}^{4,\alpha }(S^{3})\) and \(\psi \in {\mathcal{C}}^{2,\alpha }(S^{3})\). Let \(\kappa >0\) (whose value will be fixed later on), we further assume that the functions φ, ψ satisfy
Define
then we look for a solution of (28) of the form \(w=\mathbf{v}+v\), and using the fact that \(H^{i}\) is biharmonic, this amounts to solving
We fix
and denote by \({\mathscr{G}}_{\mu }\) a right inverse of \({\mathscr{L}} _{\mu }\) provided by Proposition 1. To obtain a solution of (30), it is sufficient to find \(v \in {\mathcal{C}}^{4, \alpha }_{\mu } (\mathbb{R}^{4})\) solution of
where
We denote by \({\mathscr{N}}(= {\mathscr{N}}_{\varepsilon, \lambda, \varphi,\psi })\) the nonlinear operator appearing on the right-hand side of Eq. (31), then we have the following result.
Lemma 5
Given \(\mu \in (1,2)\) and \(\kappa > 0\), there exist \(\lambda _{\kappa }> 0, \varepsilon _{\kappa }>0, c_{\kappa }>0\), and \(\bar{c}_{ \kappa }>0\) (depending on κ) such that, for all \(\lambda \in (0, \lambda _{\kappa })\) and \(\varepsilon \in (0, \varepsilon _{\kappa })\),
Moreover,
provided that \(v_{1}, v_{2} \in {\mathcal{C}}^{4, \alpha }_{\mu }( {\mathbb{R}}^{4})\) satisfy \(\| v_{i} \|_{{\mathcal{C}}^{4, \alpha } _{\mu }({\mathbb{R}}^{4})} \leqslant 2 c_{\kappa } r_{\varepsilon ,\lambda }^{2} \).
Proof
The proof of the first estimate follows from the asymptotic behavior of \(H^{i}\) together with the assumption on the norm of boundary data φ and ψ given by (29). Indeed, let \(c_{\kappa }\) be a constant depending only on κ (provided ε and λ are chosen small enough), it follows from the estimate of \(H^{i}\), given by Lemma 2, that
Since, for each \(x \in \bar{B}_{R_{\varepsilon,\lambda }}\), we have
Then, using condition \((A_{\lambda })\), we prove that \(| h(x)| \rightarrow 0\) as ε and λ tend to 0. Given \(\kappa > 0\), there exists \(c_{\kappa }> 0\) such that
On the other hand, using condition \((A_{\lambda })\), we also get
and
Making use of Proposition 1 together with (19), we get for \(\mu \in ( 1, 2)\)
To derive the second estimate, let \(v_{i}\in {\mathcal{C}}^{4, \alpha }_{\mu }({\mathbb{R}}^{4})\) satisfy \(\| v_{i} \|_{{\mathcal{C}}^{4, \alpha }_{\mu }({\mathbb{R}}^{4})} \leqslant 2 c_{\kappa } r_{ \varepsilon,\lambda }^{2}\), \(i = 1, 2\), \(\mu \in ( 1, 2)\), and condition \(A_{\lambda }\). Hence there exists \(c_{\kappa }> 0\) such that
and
So
Similarly, making use of Proposition 1 together with (19), we conclude that there exists \(\bar{c}_{\kappa }> 0\) such that
□
Reducing \(\lambda _{\kappa }>0\) and \(\varepsilon _{\kappa }>0\) if necessary, we can assume that
for all \(\lambda \in (0, \lambda _{\kappa })\) and \(\varepsilon \in (0, \varepsilon _{\kappa })\). Then (33) and (34) in Lemma 5 are enough to show that
is a contraction from
into itself and hence has a unique fixed point \(v = v (\varepsilon, \tau, \varphi, \psi; \cdot )\) in this set. This fixed point is a solution of (31) in \(\mathbb{R}^{4}\). We summarize this in the following proposition.
Proposition 4
Given \(\mu \in (1,2)\) and \(\kappa >0\), there exist \(\varepsilon _{ \kappa }>0\), \(\lambda _{\kappa }>0\), and \(c_{\kappa }>0\) (depending on κ) such that, for all \(\varepsilon \in (0, \varepsilon _{ \kappa })\), \(\lambda \in (0, \lambda _{\kappa })\) satisfying \((A_{\lambda })\), for all τ in some fixed compact subset of \([\tau _{-}, \tau ^{+}] \subset (0, \infty )\), and for given φ and ψ satisfying (26)–(29), then there exists a unique \(v\ (:= \bar{v}_{\varepsilon, \tau, \varphi, \psi })\) solution of (31) such that
solves (28) in \(\bar{B}_{R_{\varepsilon,\lambda }}\). In addition,
4 The nonlinear exterior problem
Denote \(G_{\tilde{x}} = G(x, \tilde{x})\) where G is the Green’s function given by (11) and \(H (x, \tilde{x})\) is its regular part. Clearly \(x \mapsto H(x, \tilde{x})\) is a smooth function. Let \(\tilde{\mathbf{x}} = (\tilde{x}^{j})\in \varOmega ^{m}\) close to \(\mathbf{x} = (x^{j})\), \(\tilde{\boldsymbol{\eta }} = (\tilde{\eta }^{j}) \in {\mathbb{R}}^{m}\) close to 0. Let \(\tilde{\boldsymbol{\varphi }} = ( \tilde{\varphi }^{j}) \in ({\mathcal{C}}^{4, \alpha }(S^{3}) ) ^{m}\) and \(\tilde{\boldsymbol{\psi }} = (\tilde{\psi }^{j}) \in ({\mathcal{C}} ^{2, \alpha } (S^{3}) )^{m}\) satisfy (27). We define
where \(\chi _{r_{0}}\) is a cut-off function identically equal to 1 in \(B_{r_{0}/2}\) and identically equal to 0 outside \(B_{r_{0}}\). We would like to solve the equation
with \(u = \tilde{\mathbf{u}} + \tilde{v}\) is a perturbation of \(\tilde{\mathbf{u}}\). This amounts to solving
Denote \({\varOmega _{R, \tilde{x}} = \varOmega - \bigcup_{1\leqslant j\leqslant m} B _{R}(\tilde{x}^{j})}\) for any \(R > 0\). We denote by
the extension operator defined by
It is easy to check that there exists a constant \(c = c(\nu )> 0\), only depending on ν, such that
We fix
and denote by \(\tilde{\mathscr{G}}_{\nu }\) the right inverse provided by Proposition 3. Clearly, it is enough to find \(\tilde{v} \in {\mathcal{C}}^{4, \alpha }_{\nu }(\varOmega ^{*})\) solution of
We denote by \(\tilde{\mathscr{N}}(\tilde{v})( = \tilde{\mathscr{N}} _{\varepsilon, \boldsymbol{\eta }, \tilde{\mathbf{x}}, \tilde{\boldsymbol{\varphi }}, \tilde{\boldsymbol{\psi }}}(\tilde{v})) = \tilde{\mathscr{G}}_{\nu }\circ {\tilde{\xi }}_{r_{\varepsilon, \lambda }} \circ {\tilde{\mathscr{S}}}(\tilde{v})\) the nonlinear operator on the right-hand side. Even though this is not notified in the notation, \(\tilde{\mathscr{G}}_{\nu }: {\mathcal{C}}^{0, \alpha }_{ \nu -4} (\bar{\varOmega }^{*}) \longrightarrow {\mathcal{C}}^{4, \alpha }_{\nu } (\bar{\varOmega }^{*})\) is the right inverse defined in Remark 1 with \(\bar{\varOmega }^{*} = \bar{\varOmega }- \{ \tilde{x}^{1}, \ldots, \tilde{x}^{m}\}\). Given \(\kappa >0\) (whose value will be fixed later on), we further assume that the functions \(\tilde{\varphi }^{j}\) and \(\tilde{\psi } ^{j}\) satisfy
Moreover, we assume that the parameters \(\tilde{\eta }^{j}\) and the points \(\tilde{x}^{j}\) are chosen to verify
Then the following result holds.
Lemma 6
Given \(\nu \in (-1,0)\) and \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0\) and \(c_{\kappa }>0\) (depending on κ) such that, for all \(\varepsilon \in (0,\varepsilon _{\kappa })\) and under assumptions (40) and (41), we have
and
provided that \({\tilde{v}}_{1}, {\tilde{v}}_{2} \in {\mathcal{C}}^{4, \alpha }_{\nu }(\bar{\varOmega }^{*})\) and \(\| {\tilde{v}}_{i} \|_{{\mathcal{C}} ^{4, \alpha }_{\nu }(\bar{\varOmega }^{*})} \leqslant 2c_{\kappa }r_{ \varepsilon, \lambda }^{2}\).
Proof
The proof of the first estimate follows from the asymptotic behavior of \(H^{e}\) together with the assumption on the norm of boundary data \({\tilde{\varphi }_{j}}\) and \({\tilde{\psi }_{j}}\) given by (40). Indeed, let \(c_{\kappa }\) be a constant depending only on κ (provided ε and λ are chosen small enough), it follows from the estimate of \(H^{e}\), given by Lemma 3, that
Recall that \(\tilde{\mathscr{N}} (\tilde{v}) = \tilde{\mathscr{G}} _{\nu }\circ {\tilde{\xi }}_{r_{\varepsilon },\lambda } \circ \tilde{\mathscr{S}}(\tilde{v})\), we will estimate \(\tilde{\mathscr{N}}(0)\) in different subregions of \(\bar{\varOmega }^{*}\).
• In \(B_{r_{0}}(\tilde{x}^{j})\) for \(1\leqslant j \leqslant m\), we have \(\chi _{r_{0}}(x-\tilde{x}^{j})=1\) and \(\Delta ^{2}{\tilde{\mathbf{u}}}=0\), so that
So, by an easy computation, for \(\nu \in (-1,0)\) and \(\tilde{\eta } ^{j}\) small enough, we get
• In \({\varOmega - B_{r_{0}}(\tilde{x}^{j})}\), we have \(\chi _{r_{0}}(x- \tilde{x}^{j})=0\) and \(\Delta ^{2}{\tilde{\mathbf{u}}} =0\), then
Thus
• In \(B_{r_{0}}(\tilde{x}^{j})-B_{r_{0}/2}(\tilde{x}^{j})\), using estimate (42), we have
Here
So,
Finally, making Proposition 3 with (38), we conclude that
For the proof of the second estimate, let \(\tilde{v}_{1}\) and \(\tilde{v}_{2} \in C^{4, \alpha }_{\nu }(\bar{\varOmega }^{*})\) satisfying \(\|\tilde{v}_{i}\|_{\mathcal{C}^{4,\alpha }_{\nu }} \leqslant c_{ \kappa }r^{2}_{\varepsilon, \lambda }\), so
Then, for \(\tilde{\eta }^{j}\) small enough and using estimate (38), there exists \(\bar{c}_{\kappa }> 0\) (depending on κ) such that
Then we get the second estimate. □
Reducing \(\lambda _{\kappa }>0\) and \(\varepsilon _{\kappa }>0\) if necessary, we can assume that
for all \(\lambda \in (0, \lambda _{\kappa })\) and \(\varepsilon \in (0, \varepsilon _{\kappa })\). Then (44) and (43) are enough to show that
is a contraction from
into itself and hence has a unique fixed point \({\tilde{v}} = {\tilde{v}} (\varepsilon, \tau, \varphi, \psi; \cdot )\) in this set. This fixed point is a solution of (39) in \(\mathbb{R}^{4}\). We summarize this in the following proposition.
Proposition 5
Given \(\nu \in (-1,0)\) and \(\kappa >0\), there exist \(\varepsilon _{\kappa }>0, \lambda _{\kappa }>0\), and \(c_{\kappa }>0\) (depending on κ) such that, for all \(\varepsilon \in (0, \varepsilon _{\kappa })\) and \(\lambda \in (0, \lambda _{\kappa })\), for all set of parameters \(\tilde{\eta }^{j}\) and points \(\tilde{x}^{j}\) satisfying (41), all functions \(\tilde{\varphi }^{j}\), \(\tilde{\psi }^{j}\) satisfying (27) and (40), there exists a unique ṽ \((= \tilde{v} _{\varepsilon, \boldsymbol{\eta }, \tilde{\mathbf{x}}, \tilde{\boldsymbol{\varphi }}, \tilde{\boldsymbol{\psi }}})\) solution of (39) such that
solves (37) in \(\bar{\varOmega }^{*}\). In addition,
As in the previous section, observe that the function \(\tilde{v}_{ \varepsilon, \tilde{\eta }, \tilde{x}, \tilde{\varphi }, \tilde{\psi }}\) being obtained as a fixed point for contraction mapping depends smoothly on the parameters \(\tilde{\eta }^{j} \), the points \(\tilde{x}^{j}\), and the boundary data \(\tilde{\varphi }^{j}\) and \(\tilde{\psi }^{j}\) for \(j=1, \ldots, m\). Moreover, as in the previous section, the mapping
is compact (here D is the diffeomorphism defined in Sect. 2.2). Again this follows from the fact that the equation we solve is semilinear and in (39) the right-hand side belongs to \({\mathcal{C}} ^{8, \alpha }(\bar{\varOmega }^{*} )\).
5 The nonlinear Cauchy-data matching
We will gather the results of the previous sections, keeping the notations, applying the result of Sect. 2, Sect. 3, as well as the results of Sect. 4. Assume that \(\tilde{\mathbf{x}} = (\tilde{x}^{i}) \in \varOmega ^{m}\) are given close enough to \(\mathbf{x} = (x^{i})\) such that it satisfies (41), assume also \(\boldsymbol{\tau } = (\tau ^{i}) \in [\tau ^{-},\tau ^{+} ]^{m} \subset (0, \infty )^{m}\) (the values of \(\tau ^{-}\) and \(\tau ^{+}\) will be fixed shortly). First, we consider some set of boundary data \(\boldsymbol{\varphi } = (\varphi ^{i}) \in (\mathcal{C}^{4,\alpha }(S^{3}) )^{m}\) and \(\boldsymbol{\psi } = (\psi ^{i}) \in (\mathcal{C}^{2,\alpha }(S^{3}) ) ^{m}\) satisfying (26) and (29). According to the result of Proposition 4 and provided \(\varepsilon \in (0, \varepsilon _{\kappa })\), we can find a solution of
These solutions can be decomposed (in each \(B_{r_{\varepsilon,\lambda }}(\tilde{x}^{j})\)) as
where \(R_{\varepsilon,\lambda }^{j} = \tau ^{j} r_{\varepsilon, \lambda } / \varepsilon \) and the function \(v^{j} = v_{\varepsilon, \tau ^{j}, \varphi ^{j}, \psi ^{j}} \) satisfies
Similarly, given some boundary data \(\tilde{\boldsymbol{\varphi }}: = ( \tilde{\varphi }^{i}) \in (\mathcal{C}^{4,\alpha }(S^{3}) ) ^{m}\) and \(\tilde{\boldsymbol{\psi }} = (\tilde{\psi }^{i}) \in (\mathcal{C} ^{2,\alpha }(S^{3}) )^{m}\) satisfying (27) and (40), some parameters \(\tilde{\boldsymbol{\eta }}: = ( \tilde{\eta }^{i}) \in {\mathbb{R}}^{m} \) satisfying (41), we can use the result of Proposition 5 to find a solution \(u_{\mathrm{ext}}\) of (provided \(\varepsilon \in (0, \varepsilon _{ \kappa })\))
Here the solution can be decomposed as
where the function \(\tilde{v}^{j}: ={\tilde{v}}_{\varepsilon, \tilde{\boldsymbol{\eta }}, \tilde{\mathbf{x}}, \tilde{\boldsymbol{\varphi }}, \tilde{\boldsymbol{\psi }}} \in {\mathcal{C}}^{4, \alpha }_{\nu }(\bar{ \varOmega }^{*})\) satisfies
It remains to determine the parameters and the functions in such a way that the function which is equal to \(u_{\mathrm{int}, j}\) in \(B_{r_{\varepsilon,\lambda }} (\tilde{x}^{j})\) and which is equal to \(u_{\mathrm{ext}}\) in \(\varOmega _{r_{\varepsilon,\lambda }, \tilde{x}}\) will become a smooth function. This amounts to finding the boundary data and the parameters so that, for each \(j=1, \ldots, m\),
on \(\partial B_{r_{\varepsilon,\lambda }} (\tilde{x}^{j})\). Assuming we have already (48) (for all ε small enough), the function \(u_{\varepsilon }\in {\mathcal{C}}^{4, \alpha }\) obtained by patching together the functions \(u_{\mathrm{int}, j}\) and the function \(u_{\mathrm{ext}}\) is a solution of our equation. Then the elliptic regularity theory implies that this solution is in fact smooth. This will complete the proof of our result. Because as ε tends to 0, the sequence of solutions constructed will satisfy the required properties, namely away from the points \(x^{j}\) the sequence \(u_{\varepsilon }\) converges to \(\sum_{j} G_{x^{j}}\). Before we proceed, the following remarks are important. It will be convenient to observe that the functions \(u_{\varepsilon, \tau ^{j}}\) can be expanded as
near \(\partial B_{r_{\varepsilon,\lambda }}\). Moreover, the function
which appears in the expression of \(u_{\mathrm{ext}}\), can be expanded as
near \(\partial B_{r_{\varepsilon,\lambda }}\), where we define
Next, in (48), all functions are defined on \(\partial B_{r _{\varepsilon,\lambda }} (\tilde{x}^{j})\); nevertheless, it will be convenient to solve, instead of (48), the following set of equations:
on \(S^{3}\). We decompose also
where \(\varphi ^{j}_{0}, \tilde{\varphi }_{0} \in {\mathbb{E}}_{0} = \mathbb{R}\), \(\varphi ^{j}_{1}, \tilde{\varphi }_{1}^{j}, \tilde{\psi }^{j}_{1} \in {\mathbb{E}}_{1} = \operatorname{Span} \{e_{1}, \ldots, e_{4}\}\), and \(\varphi ^{j}_{\perp }, \psi ^{j}_{\perp }, \tilde{\varphi }^{j}_{\perp }, \tilde{\psi }^{j}_{\perp } \in L^{2} (S ^{3})^{\perp }\), the subspace of functions which are orthogonal to \({\mathbb{E}}_{0}\) and \({\mathbb{E}}_{1}\). Projecting the set of equations (51) over \({\mathbb{E}} _{0}\) will yield the system
Here, and from now on, the term \({\mathcal{O}} (r_{\varepsilon, \lambda }^{2})\) depends nonlinearly on the variables \(\tau ^{\ell }, \tilde{x}^{\ell }, \varphi ^{\ell }, \psi ^{\ell }\), \(\tilde{\varphi } ^{\ell }, \tilde{\psi }^{\ell }\), but it is bounded (in the appropriate norm) by a constant (independent of ε and κ) time \(r_{\varepsilon,\lambda }^{2}\). Let us comment briefly on how these equations are obtained. These equations simply come from (51) when expansions (49) and (50) are used, together with the expressions of \(H^{i}\) and \(H^{e}\) given in Lemma 2 and Lemma 3, also estimates (46) and (47). Observe that the projection of the term \(\nabla _{x} E_{j} (\tilde{x} ^{j}, \tilde{\mathbf{x}}) \cdot y\) arising in (50), as well as the projection of its partial derivative with respect to r, over the set of constant functions is equal to 0, while its Laplacian vanishes identically. System (52) can be readily simplified into
We are now in a position to define \(\tau ^{-}\) and \(\tau ^{+}\) since, according to the above, as ε tends to 0, we expect that \(\tilde{x}^{j}\) will converge to \(x^{j}\) and that \(\tau ^{j}\) will converge to \(\tau ^{j}_{*}\) satisfying
and hence it is enough to choose \(\tau ^{-}\) and \(\tau ^{+}\) in such a way that
We now consider the \(L^{2}\)-projection of (51) 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 \({\mathbb{E}}_{1}\)
With these notations in mind, we obtain the system of equations
Again, let us comment briefly on how these equations are obtained. This time, the only important observation is that the term \(\nabla _{x} E _{j} (\tilde{x}^{j}, \tilde{\mathbf{x}}) \cdot y\) projects identically over \({\mathbb{E}}_{1}\) as well as its derivative with respect to r. System (53) simplifies into
Finally, we consider the \(L^{2}\)-projection onto \(L^{2}(S^{3})^{ \perp }\). This yields the system
Thanks to the result of Lemma 4, this last system can be re-written as
If we define the parameters \(\mathbf{t} = (t^{j}) \in \mathbb{R}^{m}\) by
then the system we have to solve reads
where, as usual, the term \({\mathcal{O}}(r_{\varepsilon,\lambda } ^{2})\) depends nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and κ) time \(r_{\varepsilon,\lambda }^{2}\), provided \(\varepsilon \in (0, \varepsilon _{\kappa })\). We claim that, provided that the degree of the mapping
from a neighborhood of \(\mathbf{x} \in \varOmega ^{m}\) to a neighborhood of 0 in \({\mathbb{E}}_{1}^{m}\) is equal to 1, this nonlinear system can be solved using Schauder’s fixed point theorem in the ball of radius \(\kappa r_{\varepsilon, \lambda }^{2}\) in the product space where the entries live, namely
and
Indeed, the nonlinear mapping which appears on the right-hand side of (55) is continuous, compact. In addition, this nonlinear mapping sends the ball of radius \(\kappa r_{\varepsilon,\lambda } ^{2}\) (for the natural product norm) into itself, provided κ is fixed large enough. In order to obtain the precise statement of our theorem, we simply observe that
where E is the functional defined by (12), then a sufficient condition for mapping (56) to have degree 1 is just that the point \(\mathbf{x} = (x^{1}, \ldots, x^{m})\) is a nondegenerate critical point of the functional E. This completes the proof of our theorem. □
References
Arioli, G., Gazzola, F., Grunau, H.C., Mitidieri, E.: A semilinear fourth order elliptic problem with exponential nonlinearity. SIAM J. Math. Anal. 36(4), 1226–1258 (2005)
Baraket, S., Bazarbacha, I., Trabelsi, M.: Singular limiting solutions to 4-dimensional elliptic problems involving exponentially dominated nonlinearity and nonlinear terms. Electron. J. Differ. Equ. 2015, 289 (2015)
Baraket, S., Bazarbacha, I., Trabelsi, N.: Construction of singular limits for 4-dimensional elliptic problems with exponentially dominated nonlinearity. Bull. Sci. Math. 31, 670–685 (2007)
Baraket, S., Dammak, M., Ouni, T., Pacard, F.: Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, 875–895 (2007)
Baraket, S., Pacard, F.: Construction of singular limits for a semilinear elliptic equation in dimension. Calc. Var. Partial Differ. Equ. 6, 1–38 (1998)
Ben Ayed, M., El Mehdi, K., Grossi, M.: Asymptotic behavior of least energy solutions of biharmonic equation in dimension four. Indiana Univ. Math. J. 5, 1723–1750 (2006)
Branson, T.: Group representations arising from Lorentz conformal geometry. J. Funct. Anal. 74, 199–293 (1987)
Branson, T.: Sharp inequality, the functional determinant and the complementary series. Trans. Am. Math. Soc. 347, 3671–3742 (1995)
Chang, S.Y.A.: On a fourth order differential operator—the Paneitz operator in conformal geometry, proceedings conference for the 70th birthday of A.P. Calderon 4, 155–165 (2000)
Chang, S.Y.A., Yang, P.: On a fourth order curvature invariant. In: Branson, T. (ed.) Contemporary Mathematics. AMS, and Arithmetic, vol. 237, pp. 9–28. AMS, Providence (1999)
Clapp, M., Munoz, C., Musso, M.: Singular limits for the bi-Laplacian operator with exponential nonlinearity in \(\mathbb{R} ^{4}\). Ann. Inst. Henri Poincaré 25, 1015–1041 (2008)
Del Pino, M., Kowalczyk, M., Musso, M.: Singular limits in Liouville type equations. Calc. Var. Partial Differ. Equ. 24(1), 47–81 (2005)
Esposito, P., Grossi, M., Pistoia, A.: On the existence of blowing-up solutions for a mean field equation. Ann. Inst. Henri Poincaré 22, 227–257 (2005)
Liouville, J.: Sur l’équation aux différences partielles \(\partial ^{2}\log \frac{\lambda }{\partial u \partial v}\pm \frac{ \lambda }{2a^{2}} = 0\). J. Math. 18, 17–72 (1853)
Malchiodi, A., Djadli, Z.: Existence of conformal metrics with constant Q-curvature. Ann. Math. 168(3), 813–858 (2008)
Mazzeo, R.: Elliptic theory of edge operators I. Commun. Partial Differ. Equ. 10(16), 1616–1664 (1991)
Melrose, R.: The Atiyah–Patodi–Singer Index Theorem. Res. Notes Math., vol. 4 (1993)
Pacard, F., Rivière, T.: Linear and nonlinear aspects of vortices: the Ginzburg Landau model. In: Progress in Nonlinear Differential Equations, vol. 39. Birkäuser, Basel (2000)
Suzuki, T.: Two dimensional Emden–Fowler equation with exponential nonlinearity. In: Nonlinear Diffusion Equations and Their Equilibrium States 3, pp. 493–512. Birkäuser, Basel (1992)
Wei, J.: Asymptotic behavior of a nonlinear fourth order eigenvalue problem. Commun. Partial Differ. Equ. 21(9–10), 1451–1467 (1996)
Wente, H.C.: Counter example to a conjecture of H. Hopf. Pac. J. Math. 121, 193–243 (1986)
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Baraket, S., Chebbi, S. & Chorfi, N. Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms. Bound Value Probl 2019, 131 (2019). https://doi.org/10.1186/s13661-019-1244-7
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DOI: https://doi.org/10.1186/s13661-019-1244-7
Keywords
- Biharmonic operator
- Nonlinear operator
- Singular limits
- Green’s function
- Nonlinear domain decomposition method