\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 51, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/51\hfil Local estimates for gradients] {Local estimates for gradients of solutions to elliptic equations with variable exponents} \author[F. Yao \hfil EJDE-2013/51\hfilneg] {Fengping Yao} % in alphabetical order \address{Fengping Yao \newline Department of Mathematics, Shanghai University, Shanghai 200444, China} \email{yfp@shu.edu.cn} \thanks{Submitted September 3, 2012. Published February 18, 2013.} \subjclass[2000]{35J60, 35J70} \keywords{Regularity; divergence; nonlinear; elliptic equation; gradient; \hfill\break\indent variable exponent; $p(x)$-Laplacian} \begin{abstract} In this article we present local $L^\infty$ estimates for the gradient of solutions to elliptic equations with variable exponents. Under proper conditions on the coefficients, we prove that $$\left| \nabla u\right|\in L^{\infty}_{loc}$$ for all weak solutions of $$\operatorname{div} (g(|\nabla u|^2,x) \nabla u )=0\quad \text{in } \Omega.$$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Uhlenbeck \cite{Uhlenbeck} obtained the interior H\"{o}lder regularity estimates for weak solutions of $$\operatorname{div} (\rho(| \nabla u |^2) \nabla u )=0\quad \text{in }\Omega, \label{1.31}$$ where $\Omega$ is an open bounded domain in $\mathbb{R}^n$ and $\rho\in C^1( [0, \infty))$ is a non-negative function satisfying the ellipticity conditions \begin{gather} K^{-1}(\xi+c)^{\frac{p}{2}-1}\leq \rho( \xi)+ 2 \rho'(\xi )\xi \leq K(\xi+c)^{\frac{p}{2}-1},\label{condition1} \\ |\rho'(\xi_1)\xi_1- \rho'(\xi_2)\xi_2|\leq K (\xi_1+\xi_2+c )^{p/2-1-\alpha}( \xi_1-\xi_2)^{\alpha}\label{condition2} \end{gather} for $c \geq 0$, $\alpha>0$ and $p \geq 2$. Especially when $\rho(t)=t^{(p-2)/2}$, \eqref{1.31} is reduced to the well-known $p$-Laplace equation. In this paper we discuss the nonlinear elliptic equation of the form $$\operatorname{div} (g(| \nabla u |^2,x) \nabla u)=0\quad \text{in }\Omega, \label{1.2}$$ where $g(\xi, x)\in C^1( [0, \infty)\times \Omega)$ satisfies the ellipticity conditions \begin{gather} C_1(\xi+c)^{\frac{p(x)}{2}-1}\leq g( \xi,x)+ 2 \xi g_{\xi}(\xi,x ) \leq C_2(\xi+c)^{\frac{p(x)}{2}-1}, \label{condition1b}\\ | \nabla_{x} g(\xi,x ) | \leq C_3 (\xi+c)^{\frac{p(x)-1}{2}} |\nabla p | | \ln (\xi+c ) | \label{condition111} \end{gather} for $c\geq 0$ and $C_1,C_2,C_3 >0$. Here $p\in W^{1,s}(\Omega)$ for some $s>n$ satisfies 10 : \int_{\Omega}|\frac{f}{\lambda}|^{p(x)}dx \leq 1 \big\}. Furthermore, we define $$W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega) : |\nabla u|\in L^{p(x)}(\Omega)\}$$ equipped with the norm $$\|u\|_{W^{1,p(x)}(\Omega)}=\|u\|_{L^{p(x)}(\Omega)} +\|\nabla u\|_{L^{p(x)}(\Omega)}.$$ By $W_{0}^{1,p(x)}(\Omega)$ we denote the closure of $C_0^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$. Actually, the $L^{p(x)}(\Omega)$, $W^{1,p(x)}(\Omega)$ and $W_{0}^{1,p(x)}(\Omega)$ spaces are Banach spaces. There have been many investigations (see for example \cite{Diening0,Diening,Diening1,Diening2,Fan2,Fan3,Harjulehto}) on properties of such variable exponent Sobolev spaces. As usual, the solutions of \eqref{1.2} are taken in a weak sense. We now state the definition of weak solutions. \begin{definition}\label{defweaksolution} \rm A function $u \in W^{1, p(x)}_{\rm loc}(\Omega)$ is a local weak solution of \eqref{1.2} in $\Omega$ if for any $\varphi \in W_0^{1,p(x)}( \Omega)$, we have $\int_{ \Omega}g(| \nabla u |^2,x) \nabla u \cdot \nabla \varphi dx=0.$ \end{definition} When $p(x)$ is a constant, many authors \cite{Byun2,Byun4,Dibene,Duzaar,Duzaar1,Kinnumen1,Iwaniec,Mingione} have studied the regularity estimates for weak solutions of quasilinear elliptic equations of $p$-Laplacian type and the general case. When $p(x)$ is not a constant, such elliptic problems \eqref{1.2} appear in mathematical models of various physical phenomena, such as the electro-rheological fluids (see, e.g., \cite{Acerbi0,Rajagopal,Ruzicka}). There have been many investigations \cite{Coscia,Fan0,Lyaghfouri} on H\"{o}lder estimates for the $p(x)$-Laplacian elliptic equation \eqref{1.2} and the more general case. Moreover, Acerbi and Mingione \cite{Acerbi1} proved that $|\mathrm{ \bf f}|^{p(x)}\in L^q_{\rm loc}(\Omega)\quad \Longrightarrow \quad |\nabla u|^{p(x)}\in L^q_{\rm loc}(\Omega) \quad \text{for } q>1$ of weak solutions of \eqref{1.2} under some assumptions. The purpose of this paper is to extend the results in \cite{Challal}, where Challal and Lyaghfouri obtained the local $L^{\infty}$ estimates of $| \nabla u|$ for the weak solutions of \eqref{pastequation}. We assume that $p(x)\in W^{1,s}(\Omega)$ for some $s>n$. Therefore, it follows from Sobolev embedding theorem that $p(x)$ is H\"{o}lder continuous with the exponent $\alpha=1-\frac{n}{s}$. Now let us state the main result of this work. \begin{theorem}\label{thm1.2} Let $u \in W^{1, p(x)}_{\rm loc}(\Omega)$ be a local weak solution of \eqref{1.2} in $\Omega$ under the assumptions \eqref{condition1}-\eqref{pxcondition1}. Then $$| \nabla u| \in L^{\infty}_{\rm loc}(\Omega).$$ Moreover, for each $\sigma>0$ and $\delta\in (0,\frac{sq(1+\sigma)}{s-q}-1)$ with a constant $q\in (n,s)$ there exists a positive constant $R_0$, depending only on $n,p_1,p_2,s,\sigma$ and $\| |\nabla u(\cdot)|^{p(\cdot)}\|_{L^1(\Omega)}$, such that, wherever $R \leq R_0$ and the ball $B_{8R} \subset \Omega$, $\sup_{B_{R/2}} | \nabla u|^{p(x)} \leq C\Big[\int\hspace{-0.38cm}- _{B_{2R}} | \nabla u|^{p(x) } dx +R^{\alpha}M^{(1+\sigma)\delta}\Big( \int\hspace{-0.38cm}- _{B_{8R}} | \nabla u|^{p(x) }+ 1 dx \Big)^{ 1+\sigma }\Big],$ where $M=\int_{B_{8R}} | \nabla u| ^{ p(x) } dx+1$ and $C$ depends on $n,p_1,p_2,s,\delta, \| p\|_{W^{1,s}(\Omega)}$. \end{theorem} \section{Proof of main result} In this section we prove Theorem \ref{thm1.2} by the approximation method. Our approach is much influenced by \cite{Acerbi1,Challal1,Challal,Wang1}. We first consider the following approximation problem $$\operatorname{div} (g(\epsilon+ | \nabla u^{\epsilon} |^2, x ) \nabla u^{\epsilon} )=0,\quad x\in B_{R'}, \epsilon \in (0, 1], \label{2.2}$$ where $B_{8R}\subset B_{R'}\subset \Omega$. It is standard that \eqref{2.2}, with the boundary condition $u^{\epsilon} = u$ on $\partial B_{R'}$, has a unique solution $u^{\epsilon}$ for fixed $\epsilon > 0$. Similarly to \cite{Challal}, we know $u^{\epsilon} \in W^{2,2}_{\rm loc}(\Omega)$. From \cite{Fan0} we can get $u \in C^{1, \mu }_{\rm loc}(\Omega)$ for some $\mu \in (0,1)$ and then have $u^{\epsilon} \in C^{1,\nu}(\overline{B}_{R'})$ for some $\nu \in (0,1)$ and $\|u^{\epsilon}\|_{C^{1,\nu}(\overline{B}_{R'})} \le C$, where $C$ is a constant independent of $\epsilon$. It follows from Ascoli-Arzel\`{a} theorem that there exists a sequence of $\{\epsilon_k\}$ converging to $0$ and satisfying $u^{\epsilon_k}\to u$ uniformly in $C^{1}(\overline{B}_{R'})$. Thus, we can get the result of Theorem \ref{thm1.2} by passing to the limit as $\epsilon_k\to 0$ in \eqref{finalresult} with $u^{\epsilon_k}$ replacing $u$. So it is sufficient to prove \eqref{finalresult}. For simplicity, we shall drop the index $\epsilon$ on $u^{\epsilon}$ in the exposition. Actually, from \eqref{2.2} we have $$\begin{split} &\big[ g( \epsilon+ | \nabla u|^2, x )\delta_{ij}+ 2 g_{\xi} ( \epsilon+ | \nabla u|^2, x)u_i u_j \big] u_{ij}+ g_{x_i}( \epsilon+ | \nabla u|^2, x)u_i \\ &=:a_{ij}u_{ij}+b_i u_i =0. \end{split}\label{2.3}$$ \begin{lemma}\label{lemma24gaij} If $g(\xi, x)\in C^1( [0, \infty)\times \Omega)$ satisfies the conditions \eqref{condition1} and \eqref{condition111}, then \begin{gather} C_4(\xi+c)^{\frac{p(x)}{2}-1}\le g( \xi, x)\leq C_5(\xi+c) ^{\frac{p(x)}{2}-1},\label{condition2b}\\ | g_\xi(\xi, x )\xi|\leq C_6(\xi+c)^{\frac{p(x)}{2}-1}\label{condition3} \end{gather} for the constants $0C_2>0,C_6>0$, and $$C_4\Big(c+ \epsilon+ | \nabla u|^2\Big)^{\frac{p(x)}{2}-1} | \xi|^2 \le a_{ij} \xi_i\xi_j \le C_5\Big( c+\epsilon+ | \nabla u|^2\Big)^{\frac{p(x)}{2}-1}| \xi|^2.$$ \end{lemma} \begin{proof} We prove only \eqref{condition2}. First, we find that $\xi^{1/2}g( \xi, x)=\int_{0}^{\xi}(t^{1/2}g(t, x))_tdt =\int_{0}^{\xi} \frac{1}{2}t^{-1/2} \big[ g(t, x)+ 2t g_t(t,x) \big] dt.$ Moreover, from \eqref{condition1} we deduce that $I_1=:\frac{C_1}{2}\int_{0}^{\xi} t^{-1/2} ( t+c)^{\frac{p(x)}{2}-1} dt \le \xi^{1/2}g( \xi,x) \le \frac{C_2 }{2}\int_{0}^{\xi} t^{-1/2} ( t+c)^{\frac{p(x)}{2}-1} dt=:I_2.$ To estimate of $I_1$ and $I_2$, we consider two cases. \textbf{Case 1: $c \leq \xi$.} We have \begin{align*} I_1 &\ge \frac{C_1}{2} \int_{0}^{\xi} ( t+c)^{\frac{p(x)-3}{2} } dt\\ &\ge \frac{C_1}{2} ( \frac{1}{\frac{p(x)-1}{2} }) [ ( \xi+c)^{\frac{p(x)-1}{2}}- c^{\frac{p(x)-1}{2}}]\\ &\ge \frac{C_1}{p(x)-1} [ ( \xi+c)^{\frac{p(x)-1}{2}}- c^{\frac{p(x)-1}{2}}]. \end{align*} Since $1< p_1 \le p(x) \le p_2$ and $c \le \frac{c+\xi}{2}$, we obtain \begin{align*} I_1 &\ge \frac{C_1}{p_2-1} \big[ 1- (\frac{1}{2})^{\frac{p(x)-1}{2}}\big] ( \xi+c)^{\frac{p(x)-1}{2}}\\ &\ge C_1'( \xi+c)^{\frac{p(x)-1}{2}}\\ &\ge C_1'( \xi+c)^{\frac{p(x)}{2}-1} \xi^{1/2}. \end{align*} Moreover, we deduce that $I_2 \le \frac{C_2 }{2}( \xi+c)^{\frac{p(x)}{2}-1} \int_{0}^{\xi} t^{-1/2} dt = C_2 ( \xi+c)^{\frac{p(x)}{2}-1} \xi^{1/2} \quad \text{for } p(x) \ge 2$ and \begin{align*} I_2 &\le \frac{C_2 }{2} \int_{0}^{\xi} t^{\frac{p(x)-1}{2}-1} dt\\ &= \frac{C_2 }{p(x)-1} \xi^{\frac{p(x)-1}{2} } \\ &\le \frac{C_2 }{p_1-1} ( \xi+c)^{\frac{p(x)-1}{2} } \\ &= \frac{C_2 }{p_1-1} ( \xi+c)^{\frac{p(x)}{2}-1 }( \xi+c)^{1/2} \quad \text{for } 1< p(x)<2, \end{align*} which implies $I_2 \leq \frac{C_2 }{p_1-1} ( \xi+c)^{\frac{p(x)}{2}-1 }( 2\xi)^{1/2} = \frac{\sqrt{2}C_2 }{p_1-1} ( \xi+c)^{\frac{p(x)}{2}-1 } \xi^{1/2}$ in view of the fact that $\xi+c \le 2\xi$. \textbf{Case 2: $c \geq \xi$.} Then we have $I_1 \ge \frac{C_1}{2} \xi^{-1/2} (\xi+c)^{-1/2} \int_{0}^{\xi} ( t+c)^{\frac{p(x)-1}{2} } dt.$ Furthermore, $I_1 \ge \frac{C_1}{2} \xi^{-1/2} ( \xi+c)^{\frac{p(x)}{2}-1} (\frac{1}{2} )^{\frac{p(x)-1}{2}} \xi \ge C_1'' ( \xi+c)^{\frac{p(x)}{2}-1} \xi^{1/2}$ since $$t+c \ge c \ge\frac{1}{2} (2c ) \ge \frac{1}{2}( \xi+c).$$ Since the result \eqref{condition2} is trivial when $c=0$. Without loss of generality we may as well assume that $c>0$. Moreover, we first have \begin{align*} I_2 &\le \frac{C_2 }{2} c^{-1/2}\int_{0}^{\xi} t^{-1/2} ( t+c)^{\frac{p(x)-1}{2}} dt \\ &\le \frac{C_2 }{2} c^{-1/2}( \xi+c)^{\frac{p(x)-1}{2}} \int_{0}^{\xi} t^{-1/2} dt, \end{align*} which implies \begin{align*} I_2 &\le C_2 c^{-1/2}( \xi+c)^{\frac{p(x)-1}{2}} \xi^{1/2} \\ &= C_2 (\frac{\xi+c}{c} )^{1/2}( \xi+c)^{\frac{p(x)}{2}-1} \xi^{1/2}\\ &\le \sqrt{2} C_2 ( \xi+c)^{\frac{p(x)}{2}-1} \xi^{1/2}. \end{align*} Thus, from Cases 1 and 2 we have $g( \xi,x) \ge \min\{ C_1', C_1''\}( \xi+c)^{\frac{p(x)}{2}-1}=:C_4 ( \xi+c)^{\frac{p(x)}{2}-1}.$ and $g( \xi,x) \le \max\{\frac{\sqrt{2}C_2 }{p_1-1}, \sqrt{2} C_2\}( \xi+c)^{\frac{p(x)}{2}-1}=:C_5 ( \xi+c)^{\frac{p(x)}{2}-1},$ which completes the proof. \end{proof} Now we denote $$\widetilde{a_{ij}}=\frac{a_{ij}}{(c+\epsilon+ | \nabla u|^2)^{p(x)/2-1}}. \label{2.4}$$ Then, from the lemma above we have $$C_4 | \xi|^2\le \widetilde{a_{ij}}\xi_i\xi_j \le C_5 | \xi|^2 \quad \text{for each } \xi \in \mathbb{R}^n.$$ \begin{lemma}\label{coraij} Let $v=(c+\epsilon+ | \nabla u|^2)^{p(x)/2}$. Then \begin{align*} \operatorname{div} ( \frac{1}{p(x)}\widetilde{a_{ij}}\cdot \nabla v) &\ge \operatorname{div} \Big( a_{ij} \cdot ( c+\epsilon+ | \nabla u|^2) \cdot\ln (c+\epsilon+ | \nabla u|^2)^{1/2} \cdot \frac{\nabla p(x)}{p(x)}\Big)\\ &=: \operatorname{div}F, \end{align*} where $|F| \le C\Big[1+(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)(1+\sigma)}{2} } \big] \cdot |\nabla p | \quad \text{for any } \sigma >0.$ \end{lemma} \begin{proof} We first find that \begin{align*} &v_{x_j}\\ &=\Big((c+\epsilon+ | \nabla u|^2)\ln (c+\epsilon+ | \nabla u|^2) ^{1/2} p_{x_j}(x)+ p(x) u_{kj} u_{k}\Big) (c+\epsilon+ | \nabla u|^2)^{\frac{p(x)}{2}-1} \end{align*} and \begin{align*} &\operatorname{div} ( \frac{1}{p(x)}\widetilde{a_{ij}}\cdot \nabla v)\\ &=\operatorname{div} ( a_{ij} \cdot(c+ \epsilon+ | \nabla u|^2) \cdot\ln (c+\epsilon+ | \nabla u|^2)^{1/2} \cdot \frac{\nabla p(x)}{p(x)})+ (a_{ij} u_{kj}u_{k} )_{x_i}. \end{align*} Moreover, differentiating \eqref{2.2} with respect to $x_k$, we have $$(a_{ij} u_{kj})_{x_i}+(b_k u_{i})_{x_i}=0,\label{aiju}$$ where $b_k$ is defined in \eqref{2.3}. Therefore, from \eqref{aiju} and Lemma \ref{lemma24gaij} we have \begin{align*} &\operatorname{div} \Big( \frac{1}{p(x)}\widetilde{a_{ij}}\cdot \nabla v\Big)\\ &= \operatorname{div} \Big( a_{ij}\cdot (c+ \epsilon+ | \nabla u|^2)\cdot\ln (c+\epsilon+ | \nabla u|^2)^{1/2} \cdot \frac{\nabla p(x)}{p(x)}\Big) + a_{ij} u_{kj}u_{ki}-(b_k u_{i})_{x_i} \\ &\ge \operatorname{div} \Big( a_{ij} \cdot(c+ \epsilon+ | \nabla u|^2)\cdot\ln (c+\epsilon+ | \nabla u|^2)^{1/2} \cdot \frac{\nabla p(x)}{p(x)}\Big)-(b_k u_{i})_{x_i}\\ &=: \operatorname{div} F, \end{align*} where $$F= a_{ij} \cdot\Big(c+ \epsilon+ | \nabla u|^2\Big)\cdot\ln \Big(c+\epsilon+ | \nabla u|^2\Big)^{1/2} \cdot \frac{\nabla p(x)}{p(x)}- b_ku_i$$ Actually, for any $\sigma>0$, we deduce that \begin{align*} |F| &\le C \big\{| a_{ij}| (c+\epsilon+ | \nabla u|^2) |\ln (c+\epsilon+ | \nabla u|^2)^{1/2}| |\nabla p | + | b_k||\nabla u |\big\}\\ &\le C\big(c+\epsilon+ | \nabla u|^2\big)^{\frac{p(x)}{2}} |\ln (c+\epsilon+ | \nabla u|^2)^{1/2}| |\nabla p |\\ &\le C\big(c+\epsilon+ | \nabla u|^2\big)^{\frac{p(x)}{2}} |\ln (c+\epsilon+ | \nabla u|^2)^{\frac{p(x)}{2}}| |\nabla p |\\ &\le C \big[1+(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)(1+\sigma)}{2} } \big] |\nabla p |. \end{align*} By \eqref{condition111}, Lemma \ref{lemma24gaij} and $$|x|| \ln x | \leq C ( 1+ |x|^{1+\sigma}) \quad \text{for any } \sigma >0,$$ we complete the proof. \end{proof} Next, we shall finish the proof of the main result. \begin{proof} Using \cite[Theorem 8.17]{GT}, we obtain $$\sup_{B_{R/2}}\Big(c+\epsilon+ | \nabla u|^2\Big)^{p(x)/2} \leq C\Big( \frac{1}{R^n} \int_{B_{2R}}(c+\epsilon + | \nabla u|^2)^{\frac{p(x)}{2}} dx+K(R)\Big),$$ where $$K(R)=R^{1-\frac{n}{q}}\Big(\int_{B_{2R}}|F|^q dx\Big)^{1/q}$$ and $q\in (n, s)$ is a positive constant. Moreover, we find that $\int_{B_{2R}}|F|^q dx \le C\big\{ \int_{B_{2R}} |\nabla p |^q dx + \int_{B_{2R}} (c+\epsilon+ | \nabla u|^2)^{\frac{p(x)(\sigma +1)q}{2}} |\nabla p |^q dx\big\}$ Furthermore, using H\"{o}lder's inequality and $p \in W^{1,s}$, we obtain \begin{align*} &\int_{B_{2R}}|F|^q dx\\ &\le C \Big(\int_{B_{2R}} |\nabla p |^{s}dx\Big)^{q/s} \Big\{ \Big( \int_{B_{2R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)sq(1+\sigma)}{2(s-q)} } dx\Big)^{ \frac{s-q}{s}} + R^{\frac{n(s-q)}{s}}\Big\} \\ &\le C \Big\{ \Big( \int_{B_{2R}}(c+\epsilon + | \nabla u|^2)^{\frac{p(x)sq(1+\sigma)}{2(s-q)} } dx\Big)^{ \frac{s-q}{s}} +R^{\frac{n(s-q)}{s}}\Big\} \\ &\le C R^{\frac{n(s-q)}{s}} \Big[ \int\hspace{-0.38cm}- _{B_{2R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)sq(1+\sigma)}{2(s-q)} } +1 dx\Big]^{ \frac{s-q}{s}}. \end{align*} From \cite[Theorem 2]{Acerbi1}, for any $\delta\in (0,\frac{sq(1+\sigma)}{s-q}-1)$ there exists a positive constant $R_0$, depending only on $n,p_1,p_2,s,\delta,\sigma$ and $\|\, |\nabla u(\cdot)|^{p(\cdot)}\|_{L^1(\Omega)}$, such that, wherever $R \leq R_0$, $\int_{B_{2R}}|F|^q dx \le C R^{\frac{n(s-q)}{s}} M^{q(1+\sigma)\delta} \Big(\int\hspace{-0.38cm}- _{B_{8R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x) }{2 } }+ 1 dx \Big)^{q(1+\sigma)},$ where $$M=\int_{B_{8R}} | \nabla u| ^{ p(x) } dx+1.$$ Thus, $$K(R)\le C R^{\alpha}M^{ (1+\sigma)\delta} \Big(\int\hspace{-0.38cm}- _{B_{8R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x) }{2 } }+ 1 dx \Big)^{ 1+\sigma },$$ where the exponent $\alpha=1-\frac{n}{s}$. Finally, we conclude that $$\begin{split} &\sup_{B_{R/2}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)}{2}} \\ &\leq C\Big[ \int\hspace{-0.38cm}- _{B_{2R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x)}{2}} dx \\ &\quad +M^{(1+\sigma)\delta} R^{\alpha} \Big(\int\hspace{-0.38cm}- _{B_{8R}}(c+\epsilon+ | \nabla u|^2)^{\frac{p(x) }{2 } }+ 1 dx \Big)^{ 1+\sigma }\Big], \end{split}\label{finalresult}$$ which completes our proof. \end{proof} \subsection*{Acknowledgements} The author wishes to thank the anonymous reviewers for their valuable suggestions that improved this article. This work is supported in part by the NSFC (11001165) and Shanghai Leading Academic Discipline Project (J50101). \begin{thebibliography}{00} \bibitem{Acerbi0} E. Acerbi, G. Mingione; \emph{Regularity results for a stationary electro-rheologicaluids}, Arch. Ration. Mech. Anal., 164(3)(2002), 213-259. \bibitem{Acerbi1} E. Acerbi, G. Mingione; \emph{Gradient estimates for the $p(x)$-Laplacean system}, J. reine angew. 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