\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 33, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/33\hfil Subelliptic $p$-Laplace type systems] {Morrey estimates for subelliptic $p$-Laplace type systems with VMO coefficients in Carnot groups} \author[H. Yu, S. Zheng \hfil EJDE-2016/33\hfilneg] {Haiyan Yu, Shenzhou Zheng} \address{Haiyan Yu \newline Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China. \newline College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, China} \email{12118381@bjtu.edu.cn} \address{Shenzhou Zheng (corresponding author)\newline Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China} \email{shzhzheng@bjtu.edu.cn} \thanks{Submitted August 16, 2015. Published January 21, 2016.} \subjclass[2010]{35H20, 35B65, 35D30} \keywords{Subelliptic p-Laplace; VMO coefficients; controllable growth; \hfill\break\indent Morrey regularity; Carnot group} \begin{abstract} In this article, we study estimates in Morrey spaces to the horizontal gradient of weak solutions for a class of quasilinear sub-elliptic systems of $p$-Laplace type with VMO coefficients under the controllable growth over Carnot group if $p$ is not too far from 2. We also show a local H\"older continuity with an optimal exponent to the solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $\mathcal{G}$ be a Carnot group of step $r\ge 2$, that is, a simply connected Lie group with Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} =\oplus_{i=1}^r V_j$ such that $[V_1, V_j ]= V_{j+1}$ for $1\le j\le r-1$ and $[V_1, V_{r}] = 0$. The homogeneous dimension of $\mathcal{G}$ is defined as $Q =\sum_{i=1}^r im_i$, where $m_i = \dim V_i$ is the topological dimension with $m_1=m$. For a family of vector fields $X=(X_{1},\,X_{2},\dots,\,X_{m})$ satisfying the H\"ormander's finite rank condition rank Lie$[X_{1},\,X_{2},\dots,\,X_{m}]=r$, we assume that each component $b_{ij}(x)$ ($i=1,\,2,\dots,m$; $j=1,2,\dots,n$) of vector-field $X_{i}=\sum_{j=1}^nb_{ij}(x)\partial_j$ is a smooth function defined in Carnot group $\mathcal{G}$ for $i=1,2,\dots,m$. Therefore, $Xu=(X_{1}u,X_{2}u,\dots,X_{m}u)$ may be called the horizontal gradient of $u$, which is understood as $X_i u=\langle X_i, \nabla u \rangle=\Sigma_{j=1}^n b_{ij}(x)\partial_j u$ for $i=1,2,\dots,m$ if $u\in C^1(\mathcal{G})$, see \cite{DoM,Fol,Lu,CiGL,Zh3}. To describe our assumptions and main results better, we first recall some relevant notations and basic facts. \begin{definition} \label{def1.1} \rm An absolutely continuous path $\gamma:[0,T]\to \mathcal{G} $ is said $X$-subunit if $$ \dot{\gamma}(t)=\sum_{i=1}^{m}c_{i}(t)X_{i}(\gamma(t)) $$ with $\sum_{i=1}^{m}c_{i}(t)\leq 1$, for almost every $t\in [0,T]$. \end{definition} With $X$-subunit in hand, we can define the Carnot-Caratheodory's metric (the C-C distance) $d_X(x,y)$ as follows, see \cite{JoX,XuZ}. $$ d_X(x,y)=\inf \{T>0: \exists \gamma:[0,T]\to \mathcal{G}\ X \text{-subunit with }\ \gamma(0)=x,\,\gamma(T)=y\}. $$ Note that these vector-fields $(X_1,\dots,X_m)$ are free up to the order $r$, then there exists a positive constant $C>0$ satisfying the following relation between the C-C distance and the Euclidean metric, see \cite{XuZ,DiZ}; \begin{equation*}\label{metric-ineq} C^{-1}|x-y|\le d_X(x,y)\le C|x-y|^{1/r}. \end{equation*} In this context, all balls centered at $x$ of radius $R$ with respect to $d_X(x,y)$ are called metric balls and denoted still by $B_{R}(x)$. The distance function $d_X(\cdot,\cdot)$ satisfies the local doubling property, that is, for $B_{2R}(x)\Subset \mathcal{G} $ there exists a positive constant $R_0$ depending on vector fields $X$ and $ \mathcal{G}$ such that for all $0 0 $. For $u\in L^p_{\rm loc}(\Omega)$, if \begin{equation}\label{Morrey} \|u\|_{L^{p,\lambda}_{X}{(\Omega)}}:=\sup_{x_0\in\Omega,\, 0-p$. If $u\in L^p_{\rm loc}(\Omega)$ satisfies \begin{equation}\label{Campanato} |u|_{\mathcal {L}^{p,\lambda}_X(\Omega)}:=\sup_{x_0\in\Omega,\, 00$, such that for a.e. $x\in \Omega$ and for any $\xi\in\mathbb{R}^{nN}$ we have $$ \nu|\xi|^2\le a_{ij}^{\alpha\beta}(x)\xi_i^\alpha\xi_j^\beta\leq L|\xi|^2; $$ \item[(H2)] $a_{ij}^{\alpha\beta}(x)\in L^{\infty}(\Omega)\bigcap VMO_X$; \item[(H3)] (Controllable growth) The inhomogeneity $B(x,u,Xu)$ satisfies $$ |B(x,u,Xu)|\leq \mu(|Xu|^{p(1-\frac{1}{\gamma})}+g(x)), $$ where \[ \gamma=\begin{cases} \frac{pQ}{Q-p}, & 1\frac {\gamma}{\gamma-1}$, and $Q$ is the homogenous dimension. \end{itemize} We say that $u\in HW^{1,p}(\Omega,\mathbb{R}^{nN})$ is a weak solution of \eqref{eq1}, if \begin{equation}\label{eq2} \int_{\Omega}\big\langle\langle A(x)Xu,Xu\rangle^{\frac{p-2}{2}}A(x)Xu, X\varphi\big\rangle\,dx =\int_{\Omega}B(x,u,Xu)\varphi\,dx, \end{equation} for all $\varphi\in HW^{1,p}_{0}(\Omega)$. Recently several studies on subelliptic PDEs arising from non-commuting vector fields have been well developed based on the H\"ormander's fundamental work \cite{Ho}; see \cite{CaG,CaDG,DiZ,DoM, Fol, NaSW, RoS, XuZ, MZZ, WaL, Zh3, ZhF15}. Many important results about the fundamental solution to subelliptic operators and the Harmonic analysis theory on stratified nilpotent Lie groups have been obtained by Folland \cite{Fol}, Rothschild-Stein \cite{RoS} and Nagel-Stein-Wainger \cite{NaSW}. These results laid a solid foundation for further investigation of subelliptic Partial Differiential Equations theory. Up to the 90s, the function theory and harmonic analysis tools on Carnot groups, such as the Sobolev embedding inequality of $X$-gradient and the isoperimetric inequality, become increasingly mature, cf. \cite{CaDG,Fol,GaN,NaSW,RoS}. In fact, such subelliptic problems have received continuous attention due to their significant applications in geometry and physics \cite{BeF, RoS}. In the case of Euclidean spaces (i.e. $m = n$, $X_i =\frac{\partial}{\partial x_i}$), it was an important observation by Uhlenbeck \cite{Uh} that there exists the interior $C^{1,\alpha}$-regularity by using the classical De Giorgi-Moser-Nash iteration to the homogeneous $p$-harmonic systems as a prototype. However, it is not true for subelliptic systems of $p$-Laplacian with $p>1$. Actually, Domokos-Manfredi in \cite{DoM,DoM1} and Domokos in \cite{DoM2,DoM3} have derived $\Gamma^{1,\alpha}$ regularity for $p$-harmonic systems only while $p$ is in a neighborhood of $2$ in Heisenberg group and in Carnot group, respectively. Very recently, Zheng-Feng \cite{Zh3, ZhF15} also got the estimates and an application of the Green functions for subelliptic A-harmonic operators, and $\Gamma^{1,\alpha}$ regularity for weak solutions to subelliptic $p$-harmonic systems under the subcritical growth with $p$ close to $2$, respectively. Notice that Fazio-Fanciullo \cite{DiF} and Dong-Niu \cite{DoN} recently established the estimates of the gradient in Morrey spaces to nonlinear subelliptic systems for $p=2$. Therefore, this is a natural thought what happens if one consider a regularity of the gradient in Morrey spaces to subelliptic A-harmonic systems. In this article, we are devoted to local Morrey regularity of the horizontal gradient to a class of subelliptic A-harmonic systems with VMO coefficients under the controllable growth. More precisely, we have the following result. \begin{theorem}\label{th1} Let $u\in HW^{1,p}(\Omega,\mathbb{R}^{N})$ is a weak solution of \eqref{eq1} with $p$ close to 2. Suppose $A(x)$ and $B(x,u,Xu)$ satisfy {\rm (H1)--(H3)}. Then $Xu\in L_{X}^{p,\lambda}(\Omega,\mathbb{R}^{nN})$; moreover, there exists a constant $C> 0$ such that for any $\Omega'\Subset \Omega$ we have \begin{equation} \|Xu\|_{L_X^{p,\lambda}(\Omega',\mathbb{R}^{nN})} \leq C(\|Xu\|_{L^p(\Omega,\mathbb{R}^{nN})}+\|g\|^{\frac{1}{p-1}}_{L_X^{q,\mu} (\Omega,\mathbb{R}^{nN})}), \end{equation} where \[ \lambda=\begin{cases} \frac{p}{p-1}\frac{\mu-q}{q}, &\frac {\gamma}{\gamma-1}2$) for $A$-harmonic systems \eqref{eq1} was essentially influenced by way of a comparison with sub-Laplacian while $p$ is close to $2$ because of the Cordes conditions. This is an important technique to consider PDEs with wild coefficients, also see \cite{CaP98}. With Theorem \ref{th1} in hand, as a direct consequence we can obtain an interior H\"older continuity of weak solutions of subelliptic systems \eqref{eq1} while $Q-n<\lambdas>1$. If for all $x_0\in \Omega$ and all $R:01$ and $0\leq \theta <1$, then there exist positive constants $\delta=\delta(b,Q,q,s)$ and $C=C(b,Q,q,r)$ such that $f\in L^{t}_{\rm loc}(\Omega)$ for any $t\in [s,s+\delta)$ and \begin{equation}\label{Reverse2} \Big\{\hbox{--}\hskip-9pt\int_{B_{\frac R 2}(x_0)}f^t\,dx\Big\}^{1/t} \leq C\Big\{\hbox{--}\hskip-9pt\int_{B_R(x_0)}f^s\,dx\Big\}^{1/s} +C\Big\{\hbox{--}\hskip-9pt\int_{B_R(x_0) }h^t\,dx\Big\}^{1/t}. \end{equation} \end{lemma} With the reverse H\"{o}lder inequality above in hand, We can obtain the following higher integrability of the horizontal gradients to systems \eqref{eq1}. \begin{lemma}[Higher integrability] \label{higher-integrability} Let $u\in HW^{1,p}(\Omega)$ be any weak solution of quasilinear subelliptic systems \eqref{eq1} with $A(x), B(x,u,Xu)$ satisfying assumptions {\rm(H1)} and {\rm (H3)}. Then, there exists a higher exponent $r: p0$ such that $B_{R}:=B_{R}(x_0)\Subset \Omega$. Let $\eta$ be a cutting-off function with $\eta\in C^{\infty}_{0}(B_R)$ such that $0\le \eta(x)\le1$, $\eta=1$ for $x\in B_{R/2}$, $\eta=0$ for $x\in \mathbb{R}^{n}\setminus{\overline{B}_{R}}$ and $|X\eta|\leq\frac{C}{R}$. Let us take a test function $\varphi=\eta^p(u-\bar u_R)$ in \eqref{eq1}, it follows from \eqref{eq2} that \begin{align*} &\int_{\Omega}\Big\langle \langle A(x)Xu,Xu\rangle^{\frac{p-2}{2}}A(x)Xu, \eta^pXu+p\eta^{p-1}(u-\bar u_{R})X\eta\Big \rangle dx\\ &=\int_{\Omega}B(x,u,Xu)\eta^p(u-\bar u_R)dx, \end{align*} By considering the uniformly ellipticity (H1) and the controllable growth, it yields \begin{equation}\label{represent} \begin{aligned} &\nu^{p/2} \int_{\Omega}\eta^p|X u|^pdx\\ &\le \int_{\Omega}\eta^p\langle A(x)Xu, Xu\rangle^{p/2} dx \\ &=- \int_{\Omega}\langle \langle A(x)Xu,Xu\rangle ^{\frac{p-2}{2}}A(x)Xu,p\eta^{p-1}(u-\bar u_{R})X\eta \rangle dx\\ &\quad +\int_{\Omega}B(x,u,Xu)\eta^p(u-\bar u_R)dx \\ &\le pL^{p/2}\int_{\Omega} |\eta X u|^{p-1}|(u-\bar u_R) X\eta|dx +\mu\int_{\Omega}\eta^p|u-\bar u_R||Xu|^{p(1-\frac{1}{\gamma})}dx \\ &\quad +\mu\int_{\Omega}\eta^p|u-\bar u_R||g(x)|dx \\ &:= pL^{p/2}I_{1}+\mu I_{2}+\mu I_{3}. \end{aligned} \end{equation} Next, we estimate $I_1, I_2$ and $I_3$. For $I_1$, using Young inequality with $\varepsilon_{1}>0$ and Sobolev inequality we have \begin{equation}\label{repre-1} \begin{aligned} I_{1}&= \int_{\Omega}|\eta X u|^{p-1}|(u-\bar u_R) X\eta|dx \\ &\leq \varepsilon_{1}\int_{\Omega}|\eta Xu|^pdx +C(\varepsilon_{1})\int_{\Omega}|(u-\bar u_R)X\eta|^pdx \\ &\leq \varepsilon_{1}\int_{B_{R}}| Xu|^pdx +\frac {C(\varepsilon_{1})}{R^p}\int_{B_{R}}|u-\bar u_R|^pdx \\ &\leq \varepsilon_{1}\int_{B_{R}}| Xu|^pdx +\frac {C(\varepsilon_{1})}{R^p} \Big(\int_{B_{R}}|Xu|^{\frac{Qp}{Q+p}}dx\Big)^{\frac{Q+p}{Q}}. \end{aligned} \end{equation} For estimating $I_2$, by Sobolev inequality and H\"older inequality, it follows that \begin{equation}\label{repre-2} \begin{aligned} I_{2}&= \int_{\Omega}\eta^p|u-\bar u_R||Xu|^{p(1-\frac{1}{\gamma})}dx \\ &\leq \Big(\int_{B_{R}}|u-u_{R}|^{\gamma}dx\Big )^{1/\gamma} \Big (\int_{B_{R}}|Xu|^pdx\Big)^{1-\frac{1}{\gamma}} \\ &\leq CR^{1+Q(\frac{1}{\gamma}-\frac{1}{p})} \Big (\int_{B_{R}}|Xu|^pdx\Big )^{\frac{1}{p}-\frac{1}{\gamma}} \Big (\int_{B_{R}}|Xu|^pdx\Big ), \end{aligned} \end{equation} where $\gamma\ge p$ is defined as the assumption H3 with $1+Q(\frac{1}{p}-\frac{1}{\gamma})\ge 0$. Similarly, to estimate $I_{3}$ we use H\"older inequality, Young inequality with $\varepsilon_{2}>0$ and Sobolev inequality; it yields \begin{align*} I_{3}&= \int_{\Omega}\eta^p|u-\bar u_R||g(x)|dx \\ &\leq \Big(\int_{B_{R}}|u-\bar u_R|^{\gamma}dx\Big)^{1/\gamma} \Big(\int_{B_{R}}|g(x)|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma}} \\ &\leq CR^{1+Q(\frac{1}{\gamma}-\frac{1}{p})} \Big(\int_{B_{R}}|Xu|^pdx\Big)^{1/p}\Big(\int_{B_{R}} |g(x)|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma}} \\ &\leq \varepsilon_{2}\int_{B_{R}}|Xu|^pdx +C(\varepsilon_{2}) R^{\frac{p}{p-1}\left(1+Q(\frac{1}{\gamma}-\frac{1}{p})\right)} \Big(\int_{B_{R}}|g(x)|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma} \cdot\frac{p}{p-1}}. \end{align*} Now let us put the estimates of $I_1,I_2,I_3$ together into \eqref{represent}, we obtain \begin{align*} &\int_{B_{R}}|\eta X u|^pdx \\ &\leq \frac {C(L,p,\varepsilon_{1})}{ R^p} \Big(\int_{B_{R}}|Xu|^{\frac{Qp}{Q+p}}dx\Big)^{\frac{Q+p}{Q}}\\ &\quad +C(\mu,\varepsilon_{2})R^{\frac{p}{p-1} \big(1+Q(\frac{1}{\gamma}-\frac{1}{p})\big)} \Big(\int_{B_{R}}|g(x)|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma} \cdot\frac{p}{p-1}}\\ &\quad +\Big\{\mu\varepsilon_{2}+pL^{p/2}\varepsilon_{1} +\mu C R^{1+Q(\frac{1}{\gamma}-\frac{1}{p})} \Big(\int_{B_{R}}|Xu|^pdx\Big )^{\frac{1}{p}-\frac{1}{\gamma}} \Big\}\Big (\int_{B_{R}}|Xu|^pdx\Big). \end{align*} Let us write \[ \vartheta=\mu\varepsilon_{2}+pL^{p/2}\varepsilon_{1} +\mu C R^{1+Q(\frac{1}{\gamma}-\frac{1}{p})} \Big (\int_{B_{R}}|Xu|^pdx\Big )^{\frac{1}{p}-\frac{1}{\gamma}}. \] Notice that from the absolute continuity of the Lebesgue integral, we have that $R^{1+Q(\frac{1}{\gamma}-\frac{1}{p})}\int_{B_R}|Xu|^p\to0$ as $R\to0$. Consequently we can take small $R>0$ such that $0<\vartheta<1$, and \begin{align*} \int_{B_{\frac{R}{2}}}|X u|^pdx &\leq \frac{C}{R^p}\Big(\int_{B_{R}}|Xu|^{\frac{Qp}{Q+p}}dx\Big)^{\frac{Q+p}{Q}}\\ &\quad +CR^{\frac{p}{p-1}\left(1+Q(\frac{1}{\gamma}-\frac{1}{p})\right)} \Big(\int_{B_{R}}|g(x)|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma} \cdot\frac{p}{p-1}}+\vartheta\int_{B_{R}}|Xu|^pdx, \end{align*} which implies \begin{equation*} \hbox{--}\hskip-9pt\int_{B_{\frac{R}{2}}}|X u|^pdx \leq C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|Xu|^{\frac{Qp}{Q+p}}dx\Big)^{\frac{Q+p}{Q}} +C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|Rg(x)|^{\gamma'}dx\Big)^{\frac{p'}{\gamma'}} +\vartheta\hbox{--}\hskip-9pt\int_{B_{R}}|Xu|^pdx, \end{equation*} with $p'=\frac p{p-1}$ and $\gamma'=\frac {\gamma}{\gamma-1}$. Therefore, we obtain \begin{align*} \Big(\hbox{--}\hskip-9pt\int_{B_{\frac{R}{2}}}|X u|^pdx\Big)^{1/p} &\leq C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|Xu|^{\frac{Qp}{Q+p}}dx\Big)^{\frac{Q+p}{pQ}} +C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}(|Rg(x)|^{\gamma'/p})^p dx\Big) ^{\frac 1p{\frac{p'}{\gamma'}}}\\ &\quad +\vartheta^{1/p} \Big(\hbox{--}\hskip-9pt\int_{B_{R}}|Xu|^pdx\Big )^{1/p}. \end{align*} Using the reverse H\"older inequality of Lemma \ref{Reverse}, it yields \begin{equation} \Big(\hbox{--}\hskip-9pt\int_{B_{\frac{R}{2}}}|X u|^rdx\Big)^{1/r} \leq C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|Xu|^pdx\Big)^{1/p} +C\Big(\hbox{--}\hskip-9pt\int_{B_{R}}(|Rg(x)|^{\gamma'/p})^r dx\Big)^{\frac 1r{\frac{p'}{\gamma'}}}, \end{equation} for some $p \frac {\gamma}{\gamma-1}$. Note that \begin{align*} \Big(\hbox{--}\hskip-9pt\int_{B_{R}}(|Rg(x)|^{\gamma'/p})^r dx\Big)^{\frac 1r{\frac{p'}{\gamma'}}} &= R^{\frac{1}{p-1}}\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|g(x)|^{\frac{\gamma}{\gamma-1}\cdot\frac{r}{p}}dx\Big)^{\frac{p}{r}\cdot\frac{\gamma-1}{\gamma}\cdot\frac{1}{p-1}}\\ &\le R^{\frac{1}{p-1}}\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|g(x)|^q dx \Big)^{\frac 1q\cdot\frac{1}{p-1}}, \end{align*} because $r\le \frac {pq(\gamma-1)}{\gamma}$, then we obtain \eqref{higher-integr} which completes the proof. \end{proof} The following elementary inequalities concerning $A(x)$ are useful to our main proof, see \cite{KZh}. \begin{lemma}\label{elementary} Suppose that $A=(A_{ij})$ is a symmetric matrix and satisfies uniform ellipticity {\rm (H1)}. Then there exists a positive constant $C=C(p,\nu,L)$ such that for $10$ such that for any $x_0\in \Omega$ we have \begin{equation}\label{perturbe esti} \int_{B_{\rho}(x_0)}|Xu|^pdx\leq C\big(\frac{\rho}{R}\big)^{Q} \int_{B_{R}(x_0)}|Xu|^pdx+C\int_{B_{R}(x_0)}|Xu-Xv|^pdx, \end{equation} \quad for all $0<\rho\beta$. Then there exist two constants $\epsilon_0=\epsilon_0(A,\alpha,\beta)$ and $C=C(A,\alpha,\beta)$ such that for any $0<\epsilon<\epsilon_0$ we have $$ \Phi(\rho)\le C\big\{\Big(\frac {\rho}R\Big)^{\beta}\Phi(R)+B\rho^{\beta}\big\}, $$ for any $0<\rho< R\leq R_0=\operatorname{dist}(x_0,\partial\Omega)$. \end{lemma} Finally, the following equivalence of spaces is useful to prove a local H\"older continuity of the weak solutions based on the main Theorem, see \cite{DoN,DiF}. \begin{lemma}\label{equivalence} If $0<\lambda0$ yield \begin{align*} &\int_{B_{R}}|A_R-A(x)||Xu|^{p-1}|Xu-Xv|dx\\ &\leq \Big(\int_{B_{R}}|A_R-A(x)|^{\frac{p}{p-1}}|Xu|^pdx\Big)^{1-\frac{1}{p}} \Big(\int_{B_R}|Xu-Xv|^p dx\Big)^{1/p}\\ &\leq C(\varepsilon_4)\int_{B_{R}}|A_R-A(x)|^{\frac{p}{p-1}}|Xu|^pdx+\varepsilon_4\int_{B_R}|Xu-Xv|^pdx\\ &\leq C(\varepsilon_4)|B_R|\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|A_R-A(x)|^{\frac{p}{p-1} \cdot\frac{r}{r-p}}dx\Big)^{1-\frac{p}{r}} \Big(\hbox{--}\hskip-9pt\int_{B_R}|Xu|^{r}dx\Big)^{p/r}\\ &\quad +\varepsilon_4\int_{B_R}|Xu-Xv|^pdx, \end{align*} where an $r>p$ is the same integrable index as that in Lemma \ref{higher-integrability}. Setting $t=\frac{pr}{(p-1)(r-p)}$, and by a higher integrability from Lemma \ref{higher-integrability} we obtain \begin{align*} J_1 &\leq C(\varepsilon_4)|B_R|\Big(\hbox{--}\hskip-9pt\int_{B_{R}}|A_R-A(x)|^tdx\Big)^{1-\frac{p}{r}} \Big(\hbox{--}\hskip-9pt\int_{B_R}|Xu|^pdx+R^{\frac{p}{p-1}} \Big(\hbox{--}\hskip-9pt\int_{B_R}|g|^q dx\Big)^{\frac{1}{q} \cdot\frac{p}{p-1}}\Big) \\ &\quad +\varepsilon_4\int_{B_R}|Xu-Xv|^pdx \\ &\leq C(\varepsilon_4)M_{A}(R)^{1-\frac{p}{r}}\int_{B_R}|Xu|^pdx +CR^{Q+\frac{p}{p-1}\cdot\frac{q-Q}{q}}\|g\|_{L^q}^{\frac{p}{p-1}}\\ &\quad +\varepsilon_4\int_{B_R}|Xu-Xv|^pdx. \end{align*} To estimate $J_2$, we employ H\"{o}der inequality and Young inequality again, and obtain \begin{align*} J_2 &\leq \mu\Big(\int_{B_R}|Xu|^p dx\Big)^{1-\frac{1}{\gamma}} \Big(\int_{B_R}|u-v|^{\gamma}dx\Big)^{1/\gamma}\\ &\leq \varepsilon_5\int_{B_R}|Xu|^pdx +C(\mu,Q,p,\varepsilon_5)\int_{B_R}|u-v|^{\gamma}dx\\ &\leq \varepsilon_5\int_{B_R}|Xu|^pdx+CR^{\gamma+Q(1-\frac{\gamma}{p})} \Big(\int_{B_R}|Xu-Xv|^pdx\Big)^{\frac{\gamma-p}{p}}\int_{B_R}|Xu-Xv|^pdx. \end{align*} Observing $\delta(R):=R^{\gamma+Q(1-\frac{\gamma}{p})} \Big(\int_{B_R}|Xu-Xv|^pdx\Big)^{\frac{\gamma-p}{p}}\to 0$ as $R\to 0$, then there holds \begin{equation} J_2\leq C\delta(R)\int_{B_R}|Xu-Xv|^pdx+\varepsilon_5\int_{B_R}|Xu|^pdx. \end{equation} To estimate $J_3$, by using H\"{o}lder inequality, Sobolev embedding inequality and Young inequality it follows that \begin{align*} J_3 &\leq \mu\Big(\int_{B_R}|g|^{\frac{\gamma}{\gamma-1}}dx\Big) ^{\frac{\gamma-1}{\gamma}} \Big(\int_{B_R}|u-v|^{\gamma}dx\Big)^{1/\gamma} \\ &\leq \Big(\int_{B_R}|g|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma}} R^{1+Q(\frac{1}{\gamma}-\frac{1}{p})}\Big(\int_{B_R}|Xu-Xv|^pdx\Big)^{1/p} \\ &\leq \varepsilon_6\int_{B_R}|Xu-Xv|^pdx+C(\varepsilon_6)R^{\frac{p}{p-1} [1+Q(\frac{1}{\gamma}-\frac{1}{p})]} \Big(\int_{B_R}|g|^{\frac{\gamma}{\gamma-1}}dx\Big)^{\frac{\gamma-1}{\gamma} \cdot\frac{p}{p-1}} \\ &\leq \varepsilon_6\int_{B_R}|Xu-Xv|^pdx +CR^{Q+\frac{p}{p-1}\cdot\frac{q-Q}{q}}\|g\|_{L^q}^{\frac{p}{p-1}}. \end{align*} Putting estimates of $J_1,J_2$ and $J_3$ together in \eqref{J123}, one deduces \begin{align*} &\nu^{p/2}\int_{B_R}|Xu-Xv|^pdx\\ &\leq C\big(\delta(R)+\varepsilon_4+\varepsilon_6\big) \int_{B_R}|Xu-Xv|^pdx+\Big(C(\varepsilon_4) M_{A}(R)^{1-\frac{p}{r}}+\varepsilon_5\Big)\int_{B_R}|Xu|^pdx\\ &\quad +CR^{Q+\frac{p}{p-1}\cdot\frac{q-Q}{q}}\|g\|_{L^q}^{\frac{p}{p-1}}. \end{align*} Therefore, by choosing arbitrary positive constants $\varepsilon_4,\varepsilon_6$ and $00$ we obtain \begin{equation} %\label{4.3} \int_{B_\rho}|Xu|^pdx \le C\big[\big(\frac{\rho}{R}\big)^Q+\varpi'\big] \int_{B_R}|Xu|^pdx+CR^{Q+\frac{p}{p-1}\cdot\frac{q-\mu}{q}}\|g\|_{L^{q,\mu}_X} ^{\frac{p}{p-1}}, \end{equation} with $\varpi'=C(\varepsilon_4)M_{A}(R)^{1-\frac{p}{r}}+\varepsilon_5+\varepsilon$. While $q\ge\mu$, it follows form Lemma \ref{iteration} as the same as Case 1 that $$ \int_{B_\rho}|Xu|^pdx \le C\big(\frac{\rho}{R}\big)^{Q-\lambda}\int_{B_R}|Xu|^pdx +C\rho^{Q-\lambda}\|g\|^{\frac{p}{p-1}}_{L^q(B_R)}, $$ for any $0<\lambda