In this section, we will complete the proof of the partial regularity results via the following lemmas. In the sequel, we always suppose that \(u\in HW^{1,m}(\Omega,{R}^{N})\) with \(m> 2\) is a weak solution to (1.1) with the assumptions of (H1)-(H4), and (μ1)-(μ3). First of all, we provide a linearization strategy for nonlinear sub-elliptic systems as in (1.1).
Lemma 4
Let
\(B_{\rho}(\xi_{0} ) \subset\subset\Omega\)
with
\(\rho\le \rho_{1}^{\frac{m}{2}} (\vert u_{0} \vert,\vert p_{0} \vert)\)
and
\(\varphi \in C_{0}^{\infty}(B_{\rho}(\xi_{0} ),{ {R}}^{N})\)
satisfying
\(|\varphi|\leq\rho^{2}\)
and
\(\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \le1\). Then there exists a constant
\(C_{1} \ge1\), such that
$$\begin{aligned} & \fint _{B_{\rho}(\xi_{0} )} {A_{i,p_{\beta}^{j} }^{\alpha}( \xi_{0} ,u_{0} ,p_{0} ) (Xu - p_{0} )X\varphi^{\alpha}\,d\xi} \\ &\quad\le C_{1} \bigl[ {\Phi(\xi_{0} , \rho,p_{0} ) + \omega^{\frac{1}{m}} \bigl( {\vert {p_{0} } \vert ,\Phi ( \xi_{0} ,\rho,p_{0} )} \bigr)\Phi ^{\frac{1}{2}} ( \xi_{0} ,\rho,p_{0} )} \\ &\qquad{} + {F\bigl(|u_{0} |,|p_{0} |\bigr) \eta^{\frac{1}{2}} \bigl( \rho^{2} \bigr)} \bigr]\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert , \end{aligned}$$
(4.1)
where
\(F(s,t) = K_{1} (s,t) +(1+s+t)^{r-1} \)
and
\(C_{1} = 2^{\frac {m-2}{2}} (C(L)+2c+ 4 + 2C_{p} ) > 1 \).
Proof
We first write
$$\bigl\vert {A_{i}^{\alpha}(\xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}(\xi_{0} ,u_{0} ,p_{0} )} \bigr\vert = \int_{0}^{1} {A_{i,p_{\beta}^{j} }^{\alpha}\bigl(\xi_{0} ,u_{0} ,\theta Xu + (1 - \theta)p_{0} \bigr)} (Xu - p_{0} )\,d\theta. $$
Noting that \(\int_{B_{\rho}(\xi_{0} )} {A_{i}^{\alpha}(\xi_{0} ,u_{0} ,p_{0} )X\varphi^{\alpha}}\,d\xi= 0\) and using the weak form of u, we conclude
$$\begin{aligned} &\fint _{B_{\rho}(\xi_{0} )} {A_{i,p_{\beta}^{j} }^{\alpha}( \xi_{0} ,u_{0} ,p_{0} ) (Xu - p_{0} )X\varphi^{\alpha}}\,d\xi \\ &\quad\leq \fint _{B_{\rho}(\xi_{0} )} { \biggl[ { \int_{0}^{1} { \bigl( {A_{i,p_{\beta}^{j} }^{\alpha}( \xi_{0} ,u_{0} ,p_{0} ) - A_{i,p_{\beta}^{j} }^{\alpha}\bigl(\xi_{0} ,u_{0} , \theta Xu + (1 - \theta)p_{0} \bigr)} \bigr) (Xu - p_{0} ) \,d\theta} } \biggr]}\,d\xi \\ &\qquad{}\times\sup_{B_{\rho}(\xi _{0} )} \vert {X\varphi} \vert \\ &\qquad{} + \fint _{B_{\rho}(\xi_{0} )} { \bigl[ {A_{i}^{\alpha}( \xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}\bigl(\xi,u_{0} + p_{0} (\xi- \xi_{0} ),Xu \bigr)} \bigr]} \sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \\ &\qquad{}+ \fint _{B_{\rho}(\xi_{0} )} { \bigl[ {A_{i}^{\alpha}\bigl( \xi,u_{0} + p_{0} (\xi- \xi_{0} ),Xu \bigr) - A_{i}^{\alpha}(\xi,u,Xu)} \bigr]} \sup _{B_{\rho}(\xi _{0} )} \vert {X\varphi} \vert \\ &\qquad{}+ \fint _{B_{\rho}(\xi_{0} )} {B^{\alpha}(\xi,u,Xu)\varphi^{\alpha}\,d\xi} \\ &\quad: = {I}' + I{I}' + II{I}' + I{V}', \end{aligned}$$
(4.2)
with the obvious meaning of I-IV. In order to estimate the term I, we use (1.4) and (1.9) to first get
$$\begin{aligned} & \bigl| {A_{i,p_{\beta}^{j} }^{\alpha}(\xi_{0} ,u_{0} ,p_{0} ) - A_{i,p_{\beta}^{j} }^{\alpha}\bigl( \xi_{0} ,u_{0} ,\theta Xu + (1 - \theta)p_{0} \bigr)} \bigr|^{\frac{1}{m} + (1 - \frac{1}{m})} \\ &\quad\le \bigl[ {C \bigl(1 + \vert {p_{0} } \vert ^{2} + \bigl\vert {\theta(Xu-p_{0}) + p_{0} } \bigr\vert ^{2} \bigr)^{\frac{m - 2}{2}}\omega \bigl(\vert {p_{0} } \vert , \bigl\vert {\theta(Xu - p_{0} )} \bigr\vert ^{2} \bigr)} \bigr]^{\frac{1}{m}} \\ &\qquad{} \times \bigl[ {L \bigl(1 + \vert {p_{0} } \vert ^{2} \bigr)^{\frac{m - 2}{2}} + L \bigl( {1 + \bigl\vert { \theta(Xu-p_{0}) + p_{0} } \bigr\vert ^{2}} \bigr)^{\frac{m - 2}{2}}} \bigr]^{1 - \frac{1}{m}} \\ &\quad\le C(L) \bigl(1 + \vert {p_{0} } \vert ^{2} \bigr)^{\frac{m - 2}{2}}\vert {Xu - p_{0} } \vert ^{m-2} \omega^{\frac{1}{m}} \bigl(\vert {p_{0} } \vert ,\vert {Xu - p_{0} } \vert ^{2} \bigr). \end{aligned}$$
(4.3)
Using (4.3), Hölder’s inequality, the fact that \(t \to \omega^{2}(s,t)\) is concave, and Jensen’s inequality, we have
$$\begin{aligned} I'&\le C(L)\fint _{B_{\rho}(\xi_{0} )} { \omega^{\frac{1}{m}} \bigl(\vert {p_{0} } \vert ,\vert {Xu - p_{0} } \vert ^{2} \bigr)} \vert {Xu - p_{0} } \vert ^{m-1}\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \,d\xi \\ &\le C(L)\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \biggl[ { \fint _{B_{\rho}(\xi_{0} )} {\omega \bigl(\vert {p_{0} } \vert , \vert {Xu - p_{0} } \vert ^{2} \bigr)}\,d\xi} \biggr]^{\frac{1}{m}} \biggl\{ {\fint _{B_{\rho}(\xi_{0} )} { {\vert {Xu - p_{0} } \vert ^{m}} }\,d\xi} \biggr\} ^{\frac{m - 1}{m}} \\ &\le C(L)\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \omega ^{\frac{1}{m}} \biggl(\vert {p_{0} } \vert ,\fint _{B_{\rho}(\xi_{0} )} {\vert {Xu - p_{0} } \vert ^{2}\,d\xi} \biggr) \biggl\{ { \fint _{B_{\rho}(\xi_{0} )} { {\vert {Xu - p_{0} } \vert ^{m}} }\,d\xi} \biggr\} ^{\frac{m - 1}{m}} \\ &\le C(L) \omega^{\frac{1}{m}} \bigl( {\vert {p_{0} } \vert , \Phi (\xi_{0} ,\rho,p_{0} )} \bigr)\Phi^{\frac{1}{2}}(\xi _{0} ,\rho,p_{0} )\sup_{B_{\rho}(\xi_{0} )}\vert {X \varphi} \vert , \end{aligned}$$
(4.4)
where we have used the fact \(\frac{m}{m-1}<2\) and the assumption \(\Phi (\xi_{0}, \rho, p_{0})\le1\).
The term \(II'\) can be estimated using the Dini continuity condition (1.6) and the fact that \(\eta(ts) \le t\eta (s)\) for \(t \ge1\). We have
$$\begin{aligned} II' \le{}& \sup_{B_{\rho}(\xi_{0} )} \vert {X \varphi} \vert \kappa( \cdot)\mu \bigl( {\rho \bigl(1 + \vert {p_{0} } \vert \bigr) } \bigr)\fint _{B_{\rho}(\xi_{0} )} {\bigl(1 + |Xu|\bigr)^{\frac{m}{2}}\,d\xi} \\ \le{}&\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \kappa( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)\eta^{\frac{1}{2}} \bigl( { \rho^{2} } \bigr)\fint _{B_{\rho}(\xi_{0} )} {\bigl(1 + |p_{0} | + |Xu - p_{0} |\bigr)^{\frac{m}{2}}\,d\xi} \\ \le{}&2^{\frac{m-2}{2}}\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \bigl[ \kappa( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{1 + \frac{m}{2}} \eta^{\frac{1}{2}} \bigl( {\rho^{2} } \bigr) + \kappa^{2} ( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{2} \eta \bigl( {\rho^{2} } \bigr) \\ &{}+ \fint _{B_{\rho}(\xi_{0} )} {|Xu - p_{0} |^{m} \,d\xi} \bigr] \\ \le{}&2^{\frac{m-2}{2}} \bigl[ {\Phi(\xi_{0} ,\rho,p_{0} ) + 2\kappa^{2} ( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{1 + \frac{m}{2}} \eta^{\frac{1}{2}} \bigl(\rho^{2} \bigr)} \bigr] \sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert , \end{aligned}$$
(4.5)
where we have used the fact that \(\eta(\rho^{2})\leq\eta^{\frac{1}{2}}(\rho^{2})\), which follows from the nondecreasing property of the function \(\eta(t)\), (η4), and our assumption \(\rho\leq\rho_{1}\leq1\).
Similarly, it follows that by using (1.6), (3.11), and the Poincaré inequality (2.4) in the special case \(p=q=2\)
$$\begin{aligned} III' \le{}&\sup_{B_{\rho}(\xi_{0} )} \vert {X \varphi } \vert \fint _{B_{\rho}(\xi_{0} )} {\kappa( \cdot)\eta^{\frac{1}{2}} \bigl( {\vert v \vert ^{2} } \bigr) \bigl(1 + |Xu|\bigr)^{\frac{m}{2}}\,d\xi} \\ \le{}& 2^{\frac{m-2}{2}}\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \biggl[ {\fint _{B_{\rho}(\xi_{0} )} {|Xu - p_{0} |^{m} \,d\xi} + \kappa^{2} ( \cdot)\fint _{B_{\rho}(\xi_{0} )} {\eta \bigl( {\vert v \vert ^{2} } \bigr)\,d\xi}} \\ &{}+ {\kappa( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{\frac{m}{2}} \fint _{B_{\rho}(\xi_{0} )} {\eta^{\frac{1}{2}} \bigl( { \vert v \vert ^{2} } \bigr)\,d\xi} } \biggr] \\ \le{}& 2^{\frac{m-2}{2}}\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \biggl[ \Phi(\xi_{0} ,\rho,p_{0} ) \\ &{} + 2\rho^{ - 2} \fint _{B_{\rho}(\xi_{0} )} {\vert v \vert ^{2}\,d\xi} + \kappa^{4} ( \cdot)\eta \bigl(\rho^{2} \bigr) + \kappa^{2} ( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{m} \eta^{\frac{1}{2}} \bigl(\rho^{2} \bigr) \biggr] \\ \le{}& 2^{\frac{m-2}{2}}\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert \biggl[ \Phi(\xi_{0} ,\rho,p_{0} ) \\ &{}+ 2C_{p} \fint _{B_{\rho}(\xi_{0} )} {\vert {Xu - p_{0} } \vert ^{2}\,d\xi} + 2\kappa^{4} ( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{m} \eta^{\frac{1}{2}} \bigl( \rho^{2} \bigr) \biggr] \\ \le{}& 2^{\frac{m-2}{2}} \bigl[ {(1 + 2C_{p} )\Phi(\xi_{0} , \rho,p_{0} ) + 2\kappa^{4} ( \cdot) \bigl(1 + \vert {p_{0} } \vert \bigr)^{m} \eta^{\frac{1}{2}} \bigl( \rho^{2} \bigr)} \bigr]\sup_{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert . \end{aligned}$$
(4.6)
Note that by the assumption \(\sup_{B_{\rho}(\xi_{0} )} \vert \varphi \vert \le\rho^{2} \le1\) and (η4), we have
$$\begin{aligned} IV' \le{}& c\fint _{B_{\rho}(\xi_{0} )} { \bigl( {1 + \vert u \vert ^{r - 1} + \vert {Xu} \vert ^{m(1 - 1/r)} } \bigr) \vert \varphi \vert \,d\xi} \\ \le{}& c\fint _{B_{\rho}(\xi_{0} )} {\vert {Xu} \vert ^{m(1 - 1/r)} \vert \varphi \vert \,d\xi} + c\fint _{B_{\rho}(\xi_{0} )} { \bigl\vert {u - u_{0} - p_{0} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert ^{r - 1} \vert \varphi \vert \,d\xi} \\ &{} + c\rho^{2} \bigl[1 + \bigl(|u_{0} | + \rho|p_{0} |\bigr)^{r - 1} \bigr] \\ \le{}& c \biggl( {\fint _{B_{\rho}(\xi_{0} )} {\vert {Xu - p_{0} } \vert ^{m} }\,d\xi} \biggr)^{\frac{{r - 1}}{r}} \biggl( { \fint _{B_{\rho}(\xi_{0} )} {\vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac{1}{r}} \\ &{} + c \biggl( {\fint _{B_{\rho}(\xi_{0} )} {\vert {p_{0} } \vert ^{m} \,d\xi } } \biggr)^{\frac{{r - 1}}{r}} \biggl( {\fint _{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac{1}{r}} \\ &{}+ c \biggl( {\fint _{B_{\rho}(\xi_{0} )} {\vert {Xu - p_{0} } \vert ^{m} }\,d\xi} \biggr)^{\frac{r-1}{m} } \biggl( { \fint _{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} }\,d\xi} \biggr)^{\frac{1}{r}} + c \rho^{2} \bigl[ {1 + \bigl( {\vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} } \bigr] \\ \leq {}& 2c \biggl( {\fint _{B_{\rho}(\xi_{0} )} {\vert {{Xu-p_{0}} } \vert ^{m} }\,d\xi} \biggr)^{ \frac{r-1}{r}} \biggl( { \fint _{B_{\rho}(\xi_{0} )} {\vert \varphi \vert ^{r} }\,d\xi} \biggr)^{\frac{1}{r}} \\ &{}+ c\rho^{2} \bigl[ {1 + \bigl( {\vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \vert {p_{0} } \vert ^{m(1 - \frac{1}{r})} } \bigr] \\ \leq {}& 2c\Phi(\xi_{0} ,\rho,p_{0} )+ 2c \eta^{\frac{1}{2}} \bigl(\rho^{2} \bigr) \bigl[ { \bigl(1+ {\vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} } \bigr], \end{aligned}$$
(4.7)
where we have used \(r-1\geq m(1-1/r)\) and \(\eta(s)\le1\) for \(s\in(0,1]\).
Combining the estimates (4.4)-(4.7) with (4.2), we obtain the conclusion with \(C_{1} = 2^{\frac{m-2}{2}} (C(L)+2c+ 4 + 2C_{p} ) \). □
The following lemma is to establish the excess improvement of the functional Φ as in (1.11). The strategy of the proof is to approximate the given solution by A-harmonic functions, for which suitable decay estimates are available from the classical theory.
Lemma 5
Assume that the conditions of Lemma
2
and the following smallness conditions hold:
$$\begin{aligned}& \omega^{\frac{1}{m}} \bigl( {\vert {u_{\xi_{0} ,\rho} } \vert + \bigl\vert {(Xu)_{\xi_{0} ,\rho} } \bigr\vert ,\Phi \bigl( \xi_{0} , \rho,(Xu)_{\xi_{0} ,\rho} \bigr)} \bigr)+\Phi^{1 / 2} \bigl( \xi_{0} ,\rho,(Xu)_{\xi_{0} ,\rho} \bigr) \le\frac{\delta}{2}, \end{aligned}$$
(4.8)
$$\begin{aligned}& C_{2}F^{2} \bigl(\vert u_{\xi_{0} , \rho} \vert,\bigl| (Xu)_{\xi_{0} , \rho}\bigr| \bigr)\eta \bigl(\rho^{2} \bigr)\leq \delta^{2}, \end{aligned}$$
(4.9)
with
\(C_{2}=8C_{1}^{2}C_{4}\), together with the radius condition
$$ \rho\le\rho^{m/2} _{1} \bigl(1+\vert u_{\xi_{0} ,\rho} \vert, 1+\bigl| (Xu)_{\xi _{0} ,\rho} \bigr| \bigr). $$
(4.10)
Then we have the following excess improvement estimate with
\(\tau\in [\gamma,1)\):
$$\Phi(\xi_{0} ,\theta\rho) \le\theta^{2\tau}\Phi( \xi_{0} ,\rho) + K^{*} \bigl(\vert {u_{\xi_{0} ,\rho} } \vert , \bigl\vert {(Xu)_{\xi_{0} , \rho} } \bigr\vert \bigr)\eta \bigl( \rho^{\frac{m}{m-1}} \bigr), $$
where we have abbreviated
\(\Phi(\xi_{0} ,r) = \Phi(\xi_{0} ,r, ( {Xu} )_{\xi_{0} ,r} )\), and
\(K^{*}(s,t) =C_{7}F^{2}(1+s,1+t)\).
Proof
We define \(w = [ {u - u_{\xi_{0} ,\rho} - (Xu)_{\xi_{0},\rho} (\xi ^{1} - \xi_{0}^{1} )} ]\sigma^{ - 1}\), where
$$ \sigma= C_{1} \sqrt{\Phi(\xi_{0} ,\rho ) + 4 \delta^{ - 2} {F^{2} \bigl(\vert {u_{\xi_{0},\rho} } \vert , \bigl\vert {(Xu)_{\xi_{0},\rho} } \bigr\vert \bigr)\eta \bigl( \rho^{2} \bigr) }} $$
(4.11)
with \(C_{1}>1\) in Lemma 4. Then we have \(Xw = \sigma^{ - 1}(Xu - (Xu)_{\xi_{0},\rho} )\). Now we consider \(B_{\rho}(\xi _{0} ) \subset\subset\Omega\) such that \(\rho\le\rho_{1}^{\frac{m}{2}} (\vert {u_{\xi_{0},\rho} } \vert ,\vert {(Xu)_{\xi_{0},\rho} } \vert )\le1\). It yields
$$ \fint _{B_{\rho}(\xi_{0} )} {\vert {Xw} \vert ^{2}}\,d \xi= \sigma^{ - 2}\Phi (\xi_{0} ,\rho) \le\frac{1}{C_{1}^{2} } \le1. $$
(4.12)
Applying Lemma 4 on \(B_{\rho}(\xi_{0} )\) to u, we have, for any \(\varphi \in C_{0}^{\infty}(B_{\rho}(\xi_{0} ),{ R}^{N})\),
$$\begin{aligned} &\fint _{B_{\rho}(\xi_{0} )} {A_{i,p_{\beta}^{j} }^{\alpha}\bigl( \xi_{0} ,u_{\xi _{0},\rho} ,(Xu)_{\xi_{0},\rho} \bigr)XwX \varphi}\,d\xi \\ &\quad\le \biggl[ {\Phi^{1 / 2}(\xi_{0} ,\rho ) + \omega^{\frac{1}{m}} \bigl( { \bigl\vert {(Xu)_{\xi_{0},\rho} } \bigr\vert , \Phi(\xi_{0} ,\rho)} \bigr) + \frac{\delta}{2}} \biggr]\sup _{B_{\rho}(\xi_{0} )} \vert {X\varphi} \vert . \end{aligned}$$
(4.13)
In consideration of the smallness condition (4.8), we see that (4.12) and (4.13) imply the conditions (3.2) and (3.3) in Lemma 2, respectively. Also note that assumptions (H1) and (H2) with \(u=u_{\xi _{0},\rho}\), and \(p=(Xu)_{\xi_{0},\rho}\), imply the conditions (3.1). So there exists a \(A_{i,p_{\beta}^{j} }^{\alpha}(\xi _{0} ,u_{\xi_{0}, \rho} , (Xu)_{\xi_{0}, \rho} )\)-harmonic function \(h \in HW^{1,2}(B_{\rho}(\xi _{0} ),{ R}^{N})\) such that
$$ \fint _{B_{\rho}(\xi_{0} )} {\vert {Xh} \vert ^{2}}\,d \xi\le1 $$
(4.14)
and
$$ \rho^{ - 2}\fint _{B_{\rho}(\xi_{0} )} {\vert {w - h} \vert ^{2}}\,d\xi \le \varepsilon. $$
(4.15)
Using Lemma 3 on the ball \(B_{2\theta\rho}(\xi_{0})\) with \(u_{0}=u_{\xi_{0}, 2\theta\rho}\), \(\theta\in(0, 1 / 4]\), and replacing \(p_{0} \) by \((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \), we obtain
$$\begin{aligned} & \int_{B_{\theta\rho} (\xi_{0} )} { \bigl( { \bigl\vert {Xu - (Xu)_{\xi_{0} , \rho} - \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr\vert ^{2} + \bigl\vert {Xu - (Xu)_{\xi_{0} , \rho} - \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr\vert ^{m} } \bigr)}\,d\xi \\ &\quad\le \frac{{C_{c} }}{{(\theta\rho)^{2} }} \int_{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl( {(Xu)_{\xi_{0} , \rho } + \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr) \bigl( \xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert } ^{2}\,d\xi \\ &\qquad{}+ \frac{{C_{c} }}{{(\theta\rho)^{m} }} \int_{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl( {(Xu)_{\xi_{0} , \rho } + \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr) \bigl( \xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert } ^{m} \,d\xi \\ &\qquad{}+ C_{c} \omega_{{G}} (2\theta\rho)^{Q} \bigl[ {K_{1} \bigl(|u_{\xi_{0} ,2\theta \rho} |,\bigl|(Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr| \bigr) \eta \bigl((2\theta\rho)^{2} \bigr) } \bigr] \\ &\qquad{}+ C_{c} \biggl[ \int_{B_{2\theta\rho}(\xi_{0})} \bigl(1+|u|^{r}+|Xu|^{m} \bigr) \,d\xi \biggr]^{m(r-1)/r(m-1)} \\ &\quad: = I'' + II'' + III''+IV''. \end{aligned}$$
(4.16)
Note that the smallness conditions (4.8)-(4.9) imply \(\sigma^{2}C_{4} (=C_{1}^{2}C_{4}\Phi+\frac{1}{2}C_{2}F^{2}\eta\delta^{-2}) \le1\) with \(C_{4} = \max\{C_{0} ,(2\theta)^{-Q}\}\), where we have assumed \(\frac {1}{2} C_{1}^{2}C_{4}\delta^{2} \leq1\), which is no restriction. Then it follows by applying the prior estimate (3.5) for the \(\mathcal{A}\)-harmonic function h
$$ \bigl\vert \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr\vert \le\sigma\sup _{B_{2\theta\rho} (\xi_{0} )} \vert {Xh} \vert \le\sigma\sqrt{C_{0} } \biggl( \fint _{B_{\rho}(\xi_{0})}|Xh|^{2}\,d\xi \biggr)^{\frac{1}{2}}\le \sigma \sqrt{C_{0} } \le1. $$
(4.17)
Furthermore, it follows by Poincaré inequality (2.4)
$$\begin{aligned} \vert u_{\xi_{0} ,2\theta\rho} \vert&\le \vert u_{\xi_{0} ,\rho} \vert+ \vert u_{\xi_{0} ,2\theta\rho} - u_{\xi_{0} ,\rho} \vert \\ &\le \vert u_{\xi_{0} ,\rho} \vert+ (2\theta)^{ - Q / 2} \biggl( { \fint _{B_{\rho}(\xi_{0} )} {\bigl\vert u - (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)} - u_{\xi_{0} ,\rho} \bigr\vert ^{2} \,d\xi} \biggr)^{1 / 2} \\ &\le \vert u_{\xi_{0} ,\rho} \vert+ (2\theta)^{ - Q / 2}\rho{C_{p} } \Phi^{1/2} (\xi_{0} ,\rho) \\ &\le\vert u_{\xi_{0} ,\rho} \vert +\frac{\sigma{C_{p}}}{C_{1}(2\theta)^{\frac{Q}{2}}} \\ &\le\vert u_{\xi_{0} ,\rho} \vert+\sigma\sqrt{C_{4}}\le\vert u_{\xi_{0} ,\rho} \vert+1, \end{aligned}$$
(4.18)
where we have used the definition of σ (4.11) and the fact \(C_{1}> {C_{p}}\).
Recall that \(g(\tau)=\int_{B_{2\theta\rho}(\xi_{0})}(u-\tau)^{2}\,d\xi\) has a minimal value at \(\tau=u_{\xi_{0},2\theta\rho}\). Noting that \(u - ( {(Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta \rho} } )(\xi^{1} - \xi_{0}^{1} )\) has mean value \(u_{\xi_{0} ,2\theta\rho} \) on the ball \(B_{2\theta\rho} (\xi_{0} )\), and using the definition of w, and Poincaré inequality (2.4), we have
$$\begin{aligned} &\frac{1}{{(\theta\rho)^{2} }}\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl( {(Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr) \bigl(\xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert ^{2}\,d\xi} \\ &\quad\le\frac{{4\sigma^{2} }}{{(2\theta\rho)^{2} }} \biggl[ {\fint _{B_{2\theta\rho} (\xi_{0} )} {\vert {w - h} \vert ^{2}\,d\xi} + \fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {h - h_{\xi_{0} ,2\theta \rho} - (Xh)_{\xi_{0} ,2\theta\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert ^{2}\,d\xi} } \biggr] \\ &\quad\le4\sigma^{2} \biggl[ {(2\theta)^{ - Q - 2} \varepsilon + C_{p}^{2} (2\theta\rho)^{2} \fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {X^{2} h} \bigr\vert ^{2}\,d\xi} } \biggr] \\ &\quad\le C_{3} \bigl(\theta^{ - Q - 2} \varepsilon + \theta^{2} \bigr) \bigl[ {\Phi (\xi_{0} ,\rho) + 4 \delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr], \end{aligned}$$
(4.19)
where \(C_{3}=C_{1}^{2}(2^{-Q}+16C_{p}^{2}C_{0})\ge1\), and in the last inequality we have used the definition of σ (4.11) with \(u_{0}=u_{\xi_{0},2\theta\rho}\), and the fact that
$$\fint _{B_{2\theta\rho}(\xi_{0})}\bigl|X^{2}h\bigr|\,d\xi\leq \sup_{B_{\rho}(\xi_{0} )}\bigl|{X^{2}h }\bigr|\leq C_{0}\rho^{-2} \fint _{B_{\rho}(\xi_{0})}|Xh|^{2} \,d\xi\leq C_{0} \rho^{-2}. $$
Then it follows
$$I''\le C_{3}C_{c} \omega_{G}(2\theta\rho)^{Q} \bigl(\theta^{ - Q - 2} \varepsilon + \theta^{2} \bigr) \bigl[ {\Phi(\xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr]. $$
For \(2< m< Q\), we have \(\frac{Q-m}{Qm}=\frac{1}{m^{\ast}} < \frac{1}{m} < \frac{1}{2} \). Then there exists \(t \in(0,1)\) such that \(\frac{1}{m} = \frac{1}{2}(1 - t) + \frac{1}{m^{\ast}} t\). Using (2.5), (2.4), Young’s inequality, and (4.19) in turn, we have
$$\begin{aligned} & \fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta \rho} \bigr) \bigl( \xi^{1} - \xi^{1}_{0} \bigr)} \bigr\vert ^{m}}\,d\xi \\ &\quad\le \biggl[ {\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl((Xu)_{\xi_{0} , \rho} + \sigma (Xh)_{\xi_{0} ,2\theta\rho} \bigr) \bigl( \xi^{1} - \xi^{1}_{0} \bigr)} \bigr\vert ^{2}}\,d\xi} \biggr]^{(1 - t)\frac{m}{2}} \\ &\qquad{} \times \biggl[ {\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {u - u_{\xi_{0} ,2\theta\rho} - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr) \bigl( \xi^{1} - \xi^{1}_{0} \bigr)} \bigr\vert ^{m^{*} }}\,d\xi} \biggr]^{t\frac{m}{m^{\ast}}} \\ &\quad\le \bigl\{ { ( {\theta\rho} )^{2}C_{3} \bigl[ \theta^{-Q-2}\varepsilon+ \theta^{2} \bigr]} { \bigl[ {\Phi( \xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr]} \bigr\} ^{(1 - t)\frac{m}{2}} \\ &\qquad{} \times \biggl( {{C_{p}^{m}}(2\theta \rho)^{m}{ \fint _{B_{2\theta\rho} (\xi_{0} )} \bigl\vert {Xu - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr)} \bigr\vert ^{m}} }\,d\xi \biggr)^{t} \\ &\quad\le \bigl( {\varepsilon^{ - t}}(2C_{p})^{mt} \bigr)^{\frac{1}{1 - t}} \bigl\{ { ( {\theta\rho} )^{2}C_{3} \bigl[\theta^{-Q-2}\varepsilon+ \theta^{2} \bigr]} \\ &\qquad{}\times{ \bigl[ { \Phi(\xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr]} \bigr\} ^{\frac{m}{2}} \\ &\qquad{}+ \varepsilon(\theta\rho)^{m}\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {Xu - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr)} \bigr\vert ^{m}}\,d\xi \\ &\quad\le \bigl( {\varepsilon^{ - t}}(2C_{p})^{mt} \bigr)^{\frac{1}{1 - t}} ( {C_{3} } )^{\frac{m}{2}} { ( {\theta\rho} )}^{m} \bigl[\theta^{-Q-2}\varepsilon+ \theta^{2} \bigr] \\ &\qquad{}\times\bigl[ {\Phi (\xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr] \\ &\qquad{}+ \varepsilon(\theta\rho)^{m}\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {Xu - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr)} \bigr\vert ^{m}}\,d\xi, \end{aligned}$$
(4.20)
here we have used the assumption \(\sigma^{2}C_{4} \le1\). Furthermore, we obtain the estimate for the term \(II''\),
$$\begin{aligned} II''\le{}&C_{c}\omega_{G} ( {2 \theta\rho} )^{Q} \bigl( {\varepsilon^{ - t}}(2C_{p})^{mt} \bigr)^{\frac{1}{1 - t}} ( {C_{3} } )^{\frac{m}{2}} \bigl[ \theta^{-Q-2}\varepsilon+ \theta ^{2} \bigr] \\ &{}\times \bigl[ {\Phi(\xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr] \\ &{}+ \varepsilon C_{c} \int_{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {Xu - \bigl((Xu)_{\xi_{0} , \rho} + \sigma(Xh)_{\xi_{0} ,2\theta\rho} \bigr)} \bigr\vert ^{m}}\,d\xi. \end{aligned}$$
(4.21)
Also, (4.17) and (4.18) yield
$$\begin{aligned} III'' \le{}& C_{c} \omega_{{G}} (2\theta\rho)^{Q} \bigl[ {K_{1} \bigl(1 + |u_{\xi _{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl((2\theta\rho)^{2} \bigr) } \bigr] \\ \le{}&C_{c} \omega_{G} (2\theta\rho)^{Q}F \bigl(1+|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl((2\theta \rho)^{2} \bigr). \end{aligned}$$
Using the Poincaré type inequality, we have
$$\begin{aligned} & \biggl[ {\fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl( {|Xu|^{m} + |u|^{r} + 1} \bigr)}\,d\xi} \biggr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\quad\le \biggl[ {2^{m - 1} \fint _{B_{2\theta\rho} (\xi_{0} )} { \bigl( {\bigl|Xu - (Xu)_{\xi_{0} ,\rho} \bigr|^{m} } \bigr)}\,d\xi} \biggr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} + \bigl( {2^{m - 1} \bigl|(Xu)_{\xi_{0} ,\rho} \bigr|^{m} } \bigr)^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\qquad{}+ \biggl[ {\fint _{B_{2\theta\rho} (\xi_{0} )} {2^{r - 1} \bigl( {\bigl|u - u_{\xi_{0} ,\rho} - (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)\bigr|^{r} } \bigr)}\,d\xi} \biggr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\qquad{}+ \bigl[ { \bigl( {1 + 2^{r - 1} \bigl|u_{\xi_{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)\bigr|^{r} } \bigr)} \bigr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\quad\le C \biggl[ (2\theta)^{ - Q}{\fint _{B_{\rho}(\xi_{0} )} { \bigl( {\bigl|Xu - (Xu)_{\xi_{0} ,\rho} \bigr|^{m} } \bigr)}\,d\xi} \biggr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} + \bigl( {2^{m - 1} \bigl|(Xu)_{\xi_{0} ,\rho} \bigr|^{m} } \bigr)^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\qquad{}+ C \biggl[ (2\theta)^{ - Q}{\fint _{B_{\rho}(\xi_{0} )} {\bigl|Xu - (Xu)_{\xi_{0} ,\rho} \bigr|^{m} }\,d\xi} \biggr]^{\frac{{r - 1}}{{m - 1}}} + \bigl[ { \bigl( {1 + 2^{r - 1} \bigl|u_{\xi_{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigr|^{r} } \bigr)} \bigr]^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\quad\le C \bigl[ {(2\theta)^{ - Q} \Phi(\xi_{0} ,\rho)} \bigr]^{\frac {{m(r - 1)}}{{r(m - 1)}}} + C \bigl( {1 + |u_{\xi_{0} ,\rho} | + \bigl|(Xu)_{\xi_{0} ,\rho} \bigr|} \bigr)^{\frac{{m(r - 1)}}{{r(m - 1)}}} \\ &\quad\le C \bigl( {2 + |u_{\xi_{0} ,\rho} | + \bigl|(Xu)_{\xi_{0} ,\rho} \bigr|} \bigr)^{\frac{{m(r - 1)}}{{r(m - 1)}}} , \end{aligned}$$
(4.22)
where in the last inequality we have used the fact \((2\theta)^{ - Q} \Phi(\xi_{0} ,\rho)\le1\), implied by the assumption \(\sigma^{2}C_{4}\le 1\) with \(C_{4} = \max\{C_{0} ,(2\theta)^{-Q}\}\). In view of \(\frac{m{(r - 1)}}{{m - 1}} + \frac{m^{2}}{{m - 1}}(1 - \frac{1}{r}) < \frac{{2m(r - 1)}}{{m - 1}}\), we have
$$ {IV}'' \le C \omega_{{G}} ( {2\theta\rho} )^{Q }F^{\frac {m}{r(m-1)}} \bigl( {1+\vert {u_{\xi_{0} ,\rho} } \vert , \bigl\vert {(Xu)_{\xi_{0} , \rho}} \bigr\vert } \bigr)\eta \bigl( {(2\theta\rho)^{\frac {m}{m-1}} } \bigr), $$
(4.23)
where we have used \(\frac{Qm(r-1)}{r(m-1)}\le Q+\frac{m}{m-1}\) and \((2\theta\rho)^{\frac{m}{m-1}}\le\eta ( {(2\theta\rho)^{\frac {m}{m-1}} } )\).
Joining the estimates of \(I''\), \(II''\), \(III''\), and \(IV''\) with (4.16), we obtain
$$\begin{aligned} & \int_{B_{\theta\rho} (\xi_{0} )} { \bigl\vert {Xu - (Xu)_{\xi_{0} , \rho} - \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr\vert ^{2}}\,d\xi \\ &\qquad{}+ (1 - C_{c}\varepsilon) \int_{B_{\theta\rho} (\xi_{0} )} { \bigl\vert {Xu - (Xu)_{\xi_{0} , \rho} - \sigma(Xh)_{\xi_{0} ,2\theta\rho} } \bigr\vert ^{m}}\,d\xi \\ &\quad\le C_{c}(C_{3})^{\frac{m}{2}} \omega_{G}(2 \theta\rho)^{Q} \bigl[ { \bigl( {(2C_{p})^{mt} \varepsilon^{ - t}} \bigr)^{\frac{1}{1 - t}} + 1} \bigr] \bigl[ \theta^{-Q-2}\varepsilon+ \theta^{2} \bigr] \\ &\qquad{}\times \bigl[ {\Phi(\xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr] \\ &\qquad{}+ C_{c} \omega_{G} (2\theta \rho)^{Q}F^{2} \bigl(|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl((2\theta \rho)^{2} \bigr) \\ &\qquad{}+ C \omega_{{G}} ( {2\theta\rho} )^{Q }F^{\frac {m}{r(m-1)}} \bigl( {1+\vert {u_{\xi_{0} ,\rho} } \vert , \bigl\vert {(Xu)_{\xi_{0} , \rho}} \bigr\vert } \bigr)\eta \bigl( {(2\theta \rho)^{\frac {m}{m-1}} } \bigr) \\ &\quad\le C_{5} \omega_{G} (2\theta\rho)^{Q} \bigl[ \theta^{-Q-2}\varepsilon+ \theta^{2} \bigr] \bigl[ { \Phi( \xi_{0} ,\rho) + 4\delta^{ - 2} {F^{2} \bigl(| {u_{\xi_{0} , \rho} } |, \bigl\vert (Xu)_{\xi_{0} , \rho} \bigr\vert \bigr)\eta \bigl(\rho^{2} \bigr) } } \bigr] \\ &\qquad{}+ C \omega_{G} (2\theta\rho)^{Q}F^{2} \bigl(1+|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl( { \rho^{\frac{m}{m-1}} } \bigr), \end{aligned}$$
where \(C_{5}=C_{c}(C_{3})^{\frac{m}{2}} [ { ( {(2C_{p})^{mt}\varepsilon^{ - t}} )^{\frac{1}{1 - t}} + 1} ]>1\).
We set \(P =(Xu)_{\xi_{0} , \rho}+\sigma(Xh)_{\xi_{0} ,2\theta\rho }\). Choosing a suitable small \(\varepsilon>0\) such that \((1 - C_{c}\varepsilon)>0\), and considering the smallness condition (4.10) (it implies \(\rho\le\rho_{1} (\vert u_{\xi_{0} , 2\theta\rho} \vert,\vert P\vert)\), see (4.17) and (4.18) above), we deduce that
$$\begin{aligned} &\Phi \bigl(\xi_{0} ,\theta\rho,(Xu)_{\xi_{0} ,\theta\rho} \bigr) \\ &\quad= \fint _{B_{\theta\rho} (\xi_{0} )} \bigl( \bigl\vert {Xu -(Xu)_{\xi _{0},\theta\rho} } \bigr\vert ^{2} + \bigl\vert {Xu -(Xu)_{\xi_{0},\theta\rho} } \bigr\vert ^{m} \bigr)\,d\xi \\ &\quad\le \fint _{B_{\theta\rho} (\xi_{0} )} { \bigl(\vert {Xu - P} \vert ^{2} + \vert {Xu - P} \vert ^{m} \bigr)\,d\xi} \\ &\quad\le C_{6} \bigl[\theta^{-Q-2}\varepsilon+ \theta^{2} \bigr] \bigl[ {\Phi \bigl(\xi_{0} , \rho,(Xu)_{\xi_{0} , \rho} \bigr) +4\delta^{ -2}\eta \bigl( \rho^{2} \bigr)F^{2} \bigl(| {u_{\xi_{0} ,\rho}}|, \bigl|{(Xu)_{\xi_{0} , \rho } }\bigr| \bigr)} \bigr] \\ &\qquad{}+ C_{6} F^{2} \bigl(1+|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl( {\rho^{\frac{m}{m-1}} } \bigr), \end{aligned}$$
(4.24)
where \(C_{6}=\frac{C_{5}2^{Q}}{ [1-C_{c}\varepsilon ]}>1\).
For a given \(\tau\in[\gamma,1)\), we now specify \(\varepsilon= \theta^{Q + 4}\), \(\theta\in(0,1 / 4]\) such that \(2C_{6}\theta^{2} \le\theta^{2\tau}\). Then we have
$$\begin{aligned} \Phi (\xi_{0} ,\theta\rho ) &\le \theta^{2\tau}\Phi( \xi_{0} ,\rho)+ \bigl(8C_{6}\theta^{2} \delta^{ - 2}+C_{6} \bigr)F^{2} \bigl(1+|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl( { \rho^{\frac{m}{m-1}} } \bigr) \\ &\le \theta^{2\tau}\Phi(\xi_{0} ,\rho)+ C_{7}F^{2} \bigl(1+|u_{\xi_{0} ,\rho} |,1 + \bigl|(Xu)_{\xi_{0} , \rho} \bigr| \bigr)\eta \bigl( { \rho^{\frac{m}{m-1}} } \bigr) \\ &:= \theta^{2\tau}\Phi(\xi_{0} ,\rho)+ K^{*} \bigl(|u_{\xi_{0} ,\rho} |,\bigl|(Xu)_{\xi_{0} ,\rho}\bigr| \bigr) \eta \bigl( { \rho^{\frac{m}{m-1}} } \bigr), \end{aligned}$$
(4.25)
where \(C_{7}=8C_{6}\theta^{2}\delta^{ - 2}+C_{6}>1\) and \(K^{*}(s,t)=C_{7}F^{2}(1+s, 1+ t)\). Then the proof of Lemma 5 is complete. □
For \(T > 0\), we find \(\Phi_{0} (T) > 0\) (depending on Q, N, λ, L, τ, and ω) such that
$$\begin{aligned}& \omega^{\frac{1}{m}} \bigl(2T,2\Phi_{0}(T) \bigr) + 2 \Phi_{0}^{\frac{1}{2}} (T) \le \frac{1}{2}\delta \quad \mbox{and} \end{aligned}$$
(4.26)
$$\begin{aligned}& 2(1+\sqrt{C_{p}}) \sqrt{\Phi_{0} (T)} \le \theta^{Q/2} \bigl(1 - \theta^{\tau}\bigr)T. \end{aligned}$$
(4.27)
With \(\Phi_{0} (T)\) from (4.26) and (4.27), we choose \(\rho_{0} (T) \in(0,1]\) (depending on Q, N, λ, L, τ, ω, η, and κ) such that
$$\begin{aligned}& \rho_{0} (T) \le\rho^{m/2} _{1} (1 + 2T,1 + 2T), \end{aligned}$$
(4.28)
$$\begin{aligned}& C_{2} F^{2} (2T,2T)\eta \bigl( \rho_{0} (T)^{2} \bigr) \le\delta^{2}, \end{aligned}$$
(4.29)
$$\begin{aligned}& K_{0} (T)\eta \bigl(\rho_{0} (T) ^{2} \bigr) \le \bigl(\theta^{2\gamma} - \theta^{2\tau } \bigr) \Phi_{0} (T) \quad\mbox{and} \end{aligned}$$
(4.30)
$$\begin{aligned}& 2(1 + C_{p} )K_{0} (T)H \bigl( \rho_{0} (T)^{2} \bigr) \le\theta^{Q} \bigl(1 - \theta^{\gamma}\bigr)^{2} \bigl(\theta^{2\gamma} - \theta^{2\tau} \bigr)T^{2}, \end{aligned}$$
(4.31)
where \(K_{0} (T): = K^{*}(2T,2T)\).
By the proof method of Lemma 5.1 in [12] and conditions (4.26)-(4.31), Lemma 6 can be proved. As we know, it is sufficient to complete the proof of Theorem 1 once we obtain Lemma 6.
Lemma 6
Assume that for some
\(T_{0} > 0\)
and
\(B_{\rho}(\xi_{0} ) \subset\subset \Omega\)
we have
-
(1)
\(\vert {u_{\xi_{0} ,\rho} } \vert + \vert { ( {Xu} )_{\xi _{0} ,\rho} } \vert \le T_{0} \);
-
(2)
\(\rho\le\rho_{0} (T_{0} )\);
-
(3)
\(\Phi(\xi_{0} ,\rho) \le\Phi_{0} (T_{0} )\).
Then the smallness conditions (4.8)-(4.10) are satisfied on the balls
\(B_{\theta^{j}\rho} (\xi_{0} )\)
for
\(j \in\mathrm{N} \cup \{0\}\). Moreover, the limit
\(\Lambda_{\xi_{0} } = \lim_{j \to\infty} ( {Xu} )_{\xi_{0} ,\theta^{j}\rho } \)
exists, and the estimate
$$\fint _{B_{\rho}(\xi_{0} )} {\vert {Xu - \Lambda_{\xi_{0} } } \vert } ^{2}\,d\xi\le C_{8} \biggl( { \biggl( {\frac{r}{\rho}} \biggr)^{2\tau}\Phi (\xi_{0} ,\rho) + H \bigl(r^{2} \bigr)} \biggr) $$
holds for
\(0 < r \le\rho\)
with a constant
\(C_{8} = C_{8} (Q,N,\lambda ,L,\tau,T_{0} )\).
Proof
The proof is very similar to the proof of Lemma 5.1 in [12]. We omit it here. □