\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 221, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{7mm}} \begin{document} \title[\hfilneg EJDE-2016/221\hfil Good Radon measure] {Good Radon measure for anisotropic problems with variable exponent} \author[I. Konat\'e, S. Ouaro \hfil EJDE-2016/221\hfilneg] {Ibrahime Konat\'e, Stanislas Ouaro} \address{Ibrahime Konat\'e \newline Laboratoire de Math\'ematiques et Informatique (LAMI), UFR. Sciences Exactes et appliqu\'ees, Universit\'e de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso} \email{ibrakonat@yahoo.fr} \address{Stanislas Ouaro \newline Laboratoire de Math\'ematiques et Informatique (LAMI), UFR. Sciences Exactes et Appliqu\'ees, Universit\'e de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso} \email{souaro@univ-ouaga.bf, ouaro@yahoo.fr} \thanks{Submitted March 28, 2016. Published August 17, 2016.} \subjclass[2010]{36J60, 35J65, 35J20, 35J25} \keywords{Generalized Lebesgue-Sobolev spaces; anisotropic Sobolev space; \hfill\break\indent weak solution; entropy solution; Dirichlet boundary condition; \hfill\break\indent bounded Radon diffuse measure; Marcinkiewicz space} \begin{abstract} We study nonlinear anisotropic problems with bounded Radon diffuse measure and variable exponent. We prove the existence and uniqueness of entropy solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We consider the anisotropic elliptic Dirichlet boundary-value problem \begin{equation}\label{e1.1} \begin{gathered} b(u)-\sum ^N_{i=1}\frac{\partial}{\partial x_i}a_i (x,\frac{\partial u}{\partial x_i})= \mu \quad \text{in }\Omega \\ u = 0 \quad\text{on }\partial \Omega, \end{gathered} \end{equation} where $\Omega$ is an open bounded domain of $\mathbb{R}^N\ (N\geq 3)$, with smooth boundary, $b:\mathbb{ R} \to \mathbb{R}$ is a continuous, surjective and non-decreasing function, with $b(0)=0$ and $\mu$ is a bounded Radon diffuse measure (this is $\mu$ does not charge the sets of zero $p_m(\cdot)$-capacity) such that $|\mu|(\Omega)> 0$. All papers concerned by problems like \eqref{e1.1} considered particular cases of data $b$ and measure $\mu$. Indeed, in \cite{Mihailescu}, the authors considered $b(\cdot)\equiv 0$, which permit them to exploit minimization technics to prove the existence of weak solutions and mini-max theory to prove that weak solutions are multiple. Using the same methods, Kon\'e et \emph{al.} (see \cite{Kone-Ouaro}) studied the problem \begin{equation}\label{prob-kone} \begin{gathered} -\sum ^N_{i=1}\frac{\partial}{\partial x_i}a_i(x,\frac{\partial u}{\partial x_i}) = \mu \quad \text{in } \Omega \\ u = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} where $\mu$ is a bounded Radon measure. Note that \eqref{prob-kone} is a particular case of \eqref{e1.1}, where $b(\cdot)\equiv 0$. Kon\'e et \emph{al.} also studied problem \eqref{prob-kone} when $\mu\in L^\infty (\Omega)$ (see \cite{Kone-Sado}) and Ouaro, when $\mu \in L^1(\Omega)$ (see \cite{Ouaro}). Ibrango and Ouaro studied the following problem (see \cite{Ibrango}) \begin{equation}\label{problem of Ibrango} \begin{gathered} -\sum ^N_{i=1}\frac{\partial}{\partial x_i}a_i(x,\frac{\partial u}{\partial x_i}) + b(u) = \mu \quad \text{in } \Omega \\ u = 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $\mu \in L^1(\Omega)$. In \cite{Ibrango}, the authors used the technic of monotone operators in Banach spaces and approximation methods to prove the existence and uniqueness of entropy solution of problem \eqref{problem of Ibrango}. In this paper, by using the same technics as in \cite{Ibrango}, we extend their result, taking into account a measure $\mu$ which is zero on the subsets of zero $p(\cdot)$-capacity (i.e., the capacity defined starting from $W^{1,p(\cdot)}_0(\Omega)$). In order to do that, we use a decomposition theorem proved by Nyanquini et \emph{al.} in \cite{Nyanquini}: every bounded Radon measure that is zero on the sets of zero $p(\cdot)$-capacity can be split in the sum of an element in $W^{-1,p'(\cdot)}(\Omega)$ (the dual space of $W^{1,p(\cdot)}_0(\Omega)$), and a function in $L^1(\Omega)$, and conversely, every bounded measure in $L^1(\Omega) + W^{-1,p'(\cdot)}(\Omega)$ is zero on the sets of zero $p(\cdot)$-capacity. Using the decomposition of measures result of Nyanquini et \emph{al.} (see \cite{Nyanquini}), we prove that there exists a unique entropy solution of \eqref{e1.1}. The proof of our result will strongly rely on the structure of the measure $\mu$, that is, $\mu$ belongs to $L^1(\Omega) + W^{-1,p'(\cdot)}(\Omega)$. Note that, since $b$ is not necessarily invertible, then, the uniqueness of the entropy solution is proved in terms of $b(u)$ which is clearly equivalent to the uniqueness of $u$ if and only if $b$ is invertible. Note that a good Radon measure for the problem \eqref{e1.1} is a Radon measure for which, the entropy solution of problem \eqref{e1.1} is unique. Many papers are related to problems involving variable exponents due to their applications to elastic mechanics, electrorheological fluids or image restoration. We denote by $\mathcal{M} _b(\Omega)$ the space of bounded Radon measures in $\Omega$, equiped with its standard norm $\|\cdot\|_{\mathcal{M} _b(\Omega)}$. Note that, if $\mu$ belongs to $\mathcal{M} _b(\Omega)$, then $|\mu|(\Omega)$ (the total variation of $\mu$) is a bounded positive measure on $\Omega$. Given $\mu \in \mathcal{M} _b(\Omega)$, we say that $\mu$ is diffuse with respect to the capacity $W_0^{1,p(\cdot)}(\Omega)$ ($p(\cdot)$-capacity for short) if $\mu (A)=0$, for every set $A$ such that $Cap_{p(\cdot)}(A,\Omega)=0$. For $A \subset \Omega$, we denote \[ S_{p(\cdot)}(A): = \{ u\in W^{1,p(\cdot)}_0(\Omega)\cap C_0(\Omega) : u = 1 \text{ on } A, u\geq 0 \text{ on } \Omega\}. \] The $p(\cdot)$-capacity of every subset $A$ with respect to $\Omega$ is defined by \[ Cap_ {p(\cdot)} (A, \Omega):= \inf _{u\in S_{p(\cdot)}(A)} \Big \{ \int _\Omega |\nabla u|^{p(x)}dx \Big \}. \] In the case $ S_{p(\cdot)}(A) = \emptyset$, we set $Cap_ {p(\cdot)} (A, \Omega) = +\infty$. The set of bounded Radon diffuse measure in the variable exponent setting is denoted by $\mathcal{M}_b^{p(\cdot)}(\Omega)$. We recall the decomposition result of bounded Radon diffuse measure proved by Nyanquini et \emph{al} (see \cite{Nyanquini}). \begin{theorem}\label{thm1} Let $p:\overline{\Omega} \to (1,+\infty)$ be a continuous function and $\mu \in \mathcal{M}_b(\Omega)$. Then, $\mu \in \mathcal{M}^{p(\cdot)}_b(\Omega)$ if and only if $\mu \in L^1(\Omega)+W^{-1,p'(\cdot)}(\Omega)$. \end{theorem} Recall that, in this paper, we assume that $\mu \in \mathcal{M}^{p_m(\cdot)}_b(\Omega)$, where $p_m(\cdot)$ is to be defined later. \begin{remark} \label{rmk1} \rm We do not have uniqueness of entropy solution if the measure $\mu$ does not belong to the space $\mathcal{M}_b^{p_m(\cdot)}$ (see Proof of uniqueness). Therefore, $\mathcal{M}_b^{p_m(\cdot)}$ is the set of good Radon measure for problem \eqref{e1.1}. \end{remark} The remaining part of this article is organized as follows: In Section 2, we introduce some preliminary results. In Section 3, we study the existence and uniqueness of entropy solution. We refer to \cite{Brezis,colana,fuana} as papers dealing with measures (including the case of variable exponents). \section{Preliminaries} We study problem \eqref{e1.1} under the following assumptions on the data. Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ $(N\geq 3)$ with smooth boundary domain $\partial \Omega$ and $\overrightarrow{p}(\cdot)=(p_1(\cdot),\dots ,p_N(\cdot))$ such that for any $i=1,\dots ,N$, $p_i(\cdot):\overline{\Omega}\to \mathbb{R}$ is a continuous function with \begin{equation}\label{eq1} 11, \end{equation} where $\frac{N}{\overline{p}}= \sum ^N_{i=1}\frac{1}{p_i^-}$. A prototype example that is covered by our assumption is the following anisotropic $\overrightarrow{p}$-harmonic problem: set \[ a_i(x,\xi)=|\xi|^{p_i(x)-2}\xi , \text{where $p_i(x)\geq 2$ for } i=1,\dots ,N. \] Then, we obtain the problem \begin{equation} \begin{gathered} b(u)-\sum ^N_{i=1}\frac{\partial}{\partial x_i} \Big( \Big|\frac{\partial u}{\partial x_i}\Big|^{p_i(x)-2} \frac{\partial u}{\partial x_i} \Big )= \mu \text{ in } \Omega \\ u= 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} which, in the particular case where $p_i=p$ for any $i=1,\dots ,N$, is the $p$-Laplace equation. We also recall in this section some definitions and basic properties of anisotropic Lebesgue and Sobolev spaces. We refer to \cite{radnla,radrep} for details and related properties. Set \[C_+(\overline{\Omega})=\big \{ p\in C(\overline{\Omega}): \min _{x\in \overline{\Omega}}p(x)>1 \ \ a.e. \ x\in \Omega \big \} \] and denote by \[ p_M(x):=\max (p_1(x),\dots ,p_N(x)) \quad \text{and} \quad p_m(x):=\min ( p_1(x),\dots ,p_N(x) ). \] For any $p\in C_+(\overline{\Omega})$, the variable exponent Lebesgue space is defined by \begin{align*} L^{p(\cdot)}(\Omega):=\Big\{& u: u \text{ is a measurable real valued function such that} \\ &\int _\Omega |u|^{p(x)}dx < \infty \Big \} , \end{align*} endowed with the so-called Luxembourg norm \[ |u|_{p(\cdot)}:= \inf \big \{ \lambda>0: \int _\Omega |\frac{u(x)}{\lambda } |^{p(x)}dx \leq 1 \big \}. \] The $p(\cdot)$-modular of the $L^{p(\cdot)}(\Omega)$ space is the mapping $\rho_{p(\cdot)}: L^{p(\cdot)}(\Omega)\to \mathbb{R}$ defined by \[ \rho_{p(\cdot)}(u):=\int _\Omega |u|^{p(x)}dx. \] For any $u\in L^{p(\cdot)}(\Omega)$, the following inequality (see \cite{Fan, Fan-Zhao}) will be used later \begin{equation}\label{eq8} \min \{ |u|^ {p^-}_{p(\cdot)}; |u|^ {p^+}_{p(\cdot)} \} \leq \rho _{p(\cdot)}(u)\leq \max \{ \ |u|^ {p^-}_{p(\cdot)}; \ |u|^ {p^+}_{p(\cdot)} \} . \end{equation} For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{q(\cdot)}(\Omega)$, with $\frac{1}{p(x)}+\frac{1}{q(x)}=1$ in $\Omega$, we have the H\"older type inequality (see \cite{Kovacik}) \begin{equation} | \int _\Omega uvdx |\leq \big( \frac{1}{p^-}+\frac{1}{q^-} \big) |u|_ {p(\cdot)}|v|_ {q(\cdot)}. \end{equation} If $\Omega$ is bounded and $p, q\in C_+(\overline{\Omega})$ such that $p(x)\leq q(x)$ for any $x\in \Omega$, then the embedding $L^{p(\cdot)}(\Omega) \hookrightarrow L^{q(\cdot)}(\Omega)$ is continuous (see \cite[Theorem 2.8]{Kovacik}). Herein, we need the anisotropic Sobolev space \[ W_0^{1,\overrightarrow{p}(\cdot)}(\Omega) :=\big\{ u\in W_0^{1,1}(\Omega): \frac{\partial u}{\partial x_i} \in L^{p_i(\cdot)}(\Omega), \, i=1,\dots ,N \big\}, \] which is a separable and reflexive Banach space (see \cite{Kovacik}) under the norm \[ \|u\|_{\overrightarrow{p}(\cdot)} =\sum _{i=1}^N |\frac{\partial u}{\partial x_i}| _{p_i(\cdot)} . \] We introduce the numbers \[ q=\frac{N(\overline{p}-1)}{N-1}, \quad q^* =\frac{N(\overline{p}-1)}{N-\overline{p}}=\frac{Nq}{N-q} , \] and define \[ P^*_-= \frac{N}{ \sum _{i=1}^N \frac{1}{p_i^-} -1}, \quad P^+_-=\max \{ p_1^-,\dots ,p_N^- \}, \quad P_{-,\infty}=\max \{ P^+_-, \ P^*_- \} . \] \begin{remark} \label{rmk2} \rm Since $\mu \in \mathcal{M}_b^{p_m(\cdot)}(\Omega)$, the Theorem \ref{thm1} implies that there exist $f\in L^1(\Omega)$ and $F\in (L^{p'_m(\cdot)}(\Omega))^N$ such that \begin{equation}\label{eq10} \mu = f - \operatorname{div}F, \end{equation} where $\frac{1}{p_m(x)} + \frac{1}{p'_m(x)} = 1$ for all $x\in \Omega$. \end{remark} We have the following embedding result (see \cite[Theorem 1]{Mihailescu}). \begin{theorem} \label{thm2.1} Assume that $\Omega \subset \mathbb{R}^N$ $(N\geq 3)$ is a bounded domain with smooth boundary. Assume also that the relation \eqref{eq6} is fillfuled. For any $q\in C(\overline{\Omega})$ verifying \[ 10$ depending on $N,p_1,\dots ,p_N$ if $\overline{p}k\}), \quad k\geq 0 \end{equation} satisfies an estimate of the form \begin{equation} \lambda _g(k)\leq Ck^{-q}, \quad \text{for some finite constant } C>0. \end{equation} We will use the following pseudo norm in $\mathcal{M}^q(\Omega)$ \begin{equation} \|g\|_{\mathcal{M}^q(\Omega)}:=\inf\{ C>0: \lambda _g(k)\leq Ck^{-q}, \quad \forall k>0\}. \end{equation} Finally, we use throughout the paper, the truncation function $T_k$, $(k>0)$ by \begin{equation} T_k(s) =\max \{-k,\min \{k;s\} \}. \end{equation} It is clear that $ \lim _{k\to \infty} T_k(s)=s$ and $|T_k(s)|=\min \{|s|;k\}$. We define $\mathcal{T}_0^{1,\overrightarrow{p}(\cdot)}(\Omega)$ as the set of measurable functions $u:\Omega \to \mathbb{R}$ such that $T_k(u)\in W^{1,\overrightarrow{p}(\cdot)}_0(\Omega)$. In the sequel, we denote $ W^{1,\overrightarrow{p}(\cdot)}_0(\Omega) = E$ to simplify notation. \section{Existence and uniqueness result} \begin{definition} \label{def3.1} \rm A measurable function $u\in \mathcal{T}_0^{1,\overrightarrow{p}(\cdot)}(\Omega)$ is an entropy solution of \eqref{e1.1} if $b(u)\in L^1(\Omega)$ and \begin{equation}\label{solution} \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u-v)dx +\int _\Omega b(u)T_k(u-v)dx \leq \int _\Omega T_k(u-v)d\mu, \end{equation} for all $v\in E\cap L^\infty (\Omega)$ and for every $k>0$. \end{definition} The existence result is as follows. \begin{theorem} \label{thm3.1} Assume \eqref{eq1}-\eqref{eq6} and \eqref{eq10} hold. Then, there exists at least one entropy solution of problem \eqref{e1.1}. \end{theorem} \begin{proof} The proof is done in three steps. \noindent\textbf{Step 1: Approximate problem.} We consider the problem \begin{equation}\label{approx-problem} \begin{gathered} -\sum ^N_{i=1}\frac{\partial}{\partial x_i}a_i \Big (x,\frac{\partial u_n}{\partial x_i}\Big )+ T_n(b(u_n))= \mu _n \quad \text{in } \Omega \\ u_n =0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $f_n=T_n(f)\in L^\infty (\Omega)$ and $\mu _n=f_n-\operatorname{div}F$. Note that $f_n \to f$ in $L^1(\Omega)$ as $n\to +\infty$, and \begin{equation} \|f_n\|_1 =\int _\Omega |f_n|dx \leq \int _\Omega |f|dx=\|f\|_1. \end{equation} \begin{definition} \label{def3.2} \rm A measurable function $u_n\in E$ is a weak solution for \eqref{approx-problem} if \begin{equation}\label{eq3.4} \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u_n}{\partial x_i} \Big ) \frac{\partial v}{\partial x_i}dx +\int _\Omega T_n(b(u_n))v\,dx = \int _\Omega f_nvdx +\int _\Omega F.\nabla v\,dx, \end{equation} for every $v\in E$. \end{definition} \begin{lemma}\label{existence-lemma} There exists at least one weak solution $u_n$ for problem \eqref{approx-problem} and \[ |b(u_n)|\leq \|\mu _n\|_\infty. \] \end{lemma} \begin{proof} We define the operators $A_n$ and $B_n$ as follows. \begin{equation} \langle A_nu, v\rangle = \langle Au,v\rangle +\int _\Omega T_n(b(u))vdx \quad \forall u, v \in E, \end{equation} where \begin{gather} \langle Au,v\rangle = \int _\Omega \sum _{i=1}^N a_i \big(x, \frac{\partial u}{\partial x_i}\big)\frac{\partial v}{\partial x_i}dx,\\ \langle B_n,v \rangle = \int _\Omega vd\mu _n. \end{gather} The operator $A_n$ is onto (see \cite[Lemma 3.1]{Ibrango} and \cite[Corollary 2.2]{Showalter}). Therefore, for $B_n \in E^*$, we can deduce the existence of a function $u_n\in E$ such that $\langle A_nu_n, v\rangle = \langle B_n,v\rangle$. Now, we show that $|b(u_n)|\leq \|\mu _n\|_\infty$. Indeed, let us denote by \[ H_\epsilon = \min \big( \frac{s^+}{\epsilon}; 1 \big) , \quad \operatorname{sign}_0^+(s) = \begin{cases} 1 & \text{if } s > 0 \\ 0 & \text{if } s \leq 0. \end{cases} \] If $\gamma$ is a maximal monotone operator defined on $\mathbb{R}$, we denote by $\gamma _0$ the main section of $\gamma$; i.e., \[ \gamma_0(s) = \begin{cases} \text{minimal absolute value of } \gamma (s) & \text{if } \gamma (s)\ \neq \emptyset \\ +\infty & \text{if } [s, +\infty)\cap D(\gamma) = \emptyset \\ -\infty & \text{if } (-\infty , s]\cap D(\gamma) = \emptyset. \end{cases} \] We remark that, as $\epsilon$ approaches 0, $H_\epsilon (s)$ apaprocahes $\operatorname{sign}_0^+(s)$. We take $v = H_\epsilon (u_n - M) $ as test function in \eqref{eq3.4}, for the weak solution $u_n$, where $M > 0$ (a constant to be chosen later), to obtain \begin{equation}\label{eq3.18} \begin{aligned} &\sum ^N_{i=1}\int _\Omega a_i \big(x,\frac{\partial u_n}{\partial x_i} \big) \frac{\partial }{\partial x_i}H_\epsilon (u_n - M)dx +\int _\Omega T_n(b(u_n))H_\epsilon (u_n - M)dx \\ &= \int _\Omega H_\epsilon (u_n - M)d\mu _n\,. \end{aligned} \end{equation} By \eqref{eq4} we have \begin{align*} &\sum ^N_{i=1}\int _\Omega a_i \big(x,\frac{\partial u_n}{\partial x_i} \big) \frac{\partial }{\partial x_i}H_\epsilon (u_n - M)dx \\ & = \frac{1}{\epsilon} \sum ^N_{i=1}\int _{ \{ \frac{(u_n - M)^+}{\epsilon}<1 \} } a_i \big (x,\frac{\partial u_n}{\partial x_i} \big ) \frac{\partial }{\partial x_i}(u_n - M)^+dx \\ & = \frac{1}{\epsilon} \sum ^N_{i=1}\int _{\{0< u_n - M < \epsilon \} } a_i \big (x,\frac{\partial u_n}{\partial x_i} \big ) \frac{\partial u_n}{\partial x_i} dx \geq 0\,. \end{align*} Then, \eqref{eq3.18} gives \[ \int _\Omega T_n(b(u_n))H_\epsilon (u_n - M)dx \leq \int _\Omega H_\epsilon (u_n - M)d\mu _n , \] which is equivalent to \[ \int _\Omega \Big ( T_n(b(u_n)) - T_n(b(M)) \Big )H_\epsilon (u_n - M)dx \leq \int _\Omega \Big ( \mu _n - T_n(b(M)) \Big )H_\epsilon (u_n - M)dx . \] We now let $\epsilon$ approach 0 in the inequality above to obtain \begin{equation}\label{eq3.19} \int _\Omega \Big ( T_n(b(u_n)) - T_n(b(M)) \Big )^+dx \leq \int _\Omega \Big ( \mu _n - T_n(b(M)) \Big ) \operatorname{sign}_0^+(u_n - M)dx . \end{equation} Choosing $M = b_0^{-1}(\|\mu _n\|_\infty)$ in the above inequality (since $b$ is surjective), we obtain \begin{equation}\label{eqadd} \begin{aligned} &\int _\Omega \Big( T_n(b(u_n)) - T_n(\|\mu _n\|_\infty) \Big )^+dx \\ &\leq \int _\Omega \Big ( \mu _n - T_n(\|\mu _n\|_\infty) \Big ) \operatorname{sign}_0^+(u_n - b_0^{-1}(\|\mu _n\|_\infty))dx . \end{aligned} \end{equation} For any $n \geq \|\mu _n\|_\infty$, we have \begin{align*} &\int _\Omega \Big ( \mu _n - T_n(\|\mu _n\|_\infty) \Big ) \operatorname{sign}_0^+(u_n - b_0^{-1}(\|\mu _n\|_\infty))dx \\ & = \int _\Omega \Big ( \mu _n - \|\mu _n\|_\infty \Big ) \operatorname{sign}_0^+(u_n - b_0^{-1}(\|\mu _n\|_\infty))dx \leq 0 . \end{align*} Then, \eqref{eqadd} gives \[ \int _\Omega \Big ( T_n(b(u_n)) - \|\mu _n\|_\infty \Big )^+dx \leq 0. \] Hence, for all $n > \|\mu _n\|_\infty$, we have $\Big ( T_n(b(u_n)) - \|\mu _n\|_\infty \Big )^+ = 0$ a.e. in $\Omega$, which is equivalent to \begin{equation}\label{eq3.21} T_n(b(u_n))\leq \|\mu _n\|_\infty ,\quad \text{for all } n > \|\mu _n\|_\infty. \end{equation} Let us remark that as $u_n$ is a weak solution of \eqref{eq3.4}, then $(-u_n)$ is a weak solution of the problem \begin{equation} \label{tildePn} %(\tilde{P}_n) \begin{gathered} -\sum ^N_{i=1}\frac{\partial}{\partial x_i}\tilde{a} _i\Big (x, \frac{\partial u_n}{\partial x_i}\Big )+ T_n(\tilde{b}(u_n)) = \tilde{\mu} _n \quad \text{in } \Omega \\ u_n =0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $\tilde{a}_i(x, \xi) = -a_i(x, -\xi)$, $\tilde{b}(s) = -b(-s)$ and $\tilde{\mu} _n = -\mu _n$. From \eqref{eq3.21} we deduce that \[ T_n(-b(u_n))\leq \|\mu _n\|_\infty, \quad \text{for all } n > \|\mu _n\|_\infty . \] Therefore, \begin{equation}\label{eq3.23} T_n(b(u_n)) \geq -\|\mu _n\|_\infty, \quad \text{for all } n > \|\mu _n\|_\infty . \end{equation} It follows from \eqref{eq3.21} and \eqref{eq3.23} that for all $ n > \|\mu _n\|_\infty$, $|T_n(b(u_n))|\leq \|\mu _n\|_\infty$ which implies $|b(u_n)|\leq \|\mu _n\|_\infty$ a.e. in $\Omega$. \end{proof} We now consider the problem \begin{equation}\label{approximated problem} \begin{gathered} -\sum ^N_{i=1}\frac{\partial}{\partial x_i}a_i \Big (x,\frac{\partial u_n}{\partial x_i}\Big )+ b(u_n) = \mu _n \quad \text{in } \Omega \\ u_n =0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $f_n=T_n(f)\in L^\infty (\Omega)$ and $\mu _n\in L^\infty (\Omega)$. It follows from Lemma \ref{existence-lemma} that there exists $u_n\in E$ with $b(u_n)\in L^\infty (\Omega)$ such that \begin{equation}\label{approximated solution} \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u_n}{\partial x_i} \Big ) \frac{\partial v}{\partial x_i}dx +\int _\Omega b(u_n)vdx = \int _\Omega f_nvdx +\int _\Omega F.\nabla vdx, \end{equation} for every $v\in E$. Our aim is to prove that these approximated solutions $u_n$ tend, as $n$ approaches infinity, to a measurable function $u$ which is an entropy solution of the problem $\eqref{e1.1}$. To start with, we establish some a priori estimates. \smallskip \noindent\textbf{Step 2: A priori estimates.} \begin{lemma} \label{lem3.2} There exists a positive constant $C_5$ which does not depend on $n$, such that \begin{equation} \sum _{i=1}^N\int _{\{|u_n|\leq k\}}|\frac{\partial u_n}{\partial x_i} |^{p_i^-}dx \leq C_5(k+1), \end{equation} for every $k > 0$. \end{lemma} \begin{proof} We take $v = T_k(u_n)$ as test function in \eqref{approximated solution} to obtain \begin{align*} & \sum _{i=1}^N\int _{\{|u_n|\leq k \} }a_i \Big (x,\frac{\partial u_n}{\partial x_i}\Big ) \frac{\partial u_n}{\partial x_i}dx + \int _\Omega b(u_n)T_k(u_n)dx \\ &= \int _\Omega f_nT_k(u_n)dx + \int _\Omega F\nabla T_k(u_n)dx. \end{align*} Using relation \eqref{eq4} and the fact that $ \int _\Omega b(u_n)T_k(u_n)dx \geq 0$, we obtain \begin{equation} c_3\sum _{i=1}^N\int _{\{|u_n|\leq k\}}| \frac{\partial u_n}{\partial x_i} |^{p_i(x)}dx \leq \big| \int _ \Omega f_nT_k(u_n)dx + \int _\Omega F\nabla T_k(u_n)dx \big|. \end{equation} Since \[ \big| \int _ \Omega f_nT_k(u_n)dx + \int _\Omega F\nabla T_k(u_n)dx \big| =\big| \int _\Omega T_k(u_n)d\mu _n \big| \leq k|\mu|(\Omega) \leq Ck , \] we deduce that \begin{equation}\label{eq3.10} c_3\sum _{i=1}^N\int _{\{|u_n|\leq k\}} |\frac{\partial u_n}{\partial x_i} |^{p_i(x)}dx \leq Ck. \end{equation} We have \begin{align*} &\sum _{i=1}^N \int _{\{|u_n|\leq k\}}|\frac{\partial u_n}{\partial x_i} |^{p_i^-}dx\\ &= \sum _{i=1}^N\int _{\{|u_n|\leq k; |\frac{\partial u_n}{\partial x_i}|> 1\}} |\frac{\partial u_n}{\partial x_i} |^{p^-_i}dx + \sum _{i=1}^N\int _{\{|u_n|\leq k; |\frac{\partial u_n}{\partial x_i}|\leq 1 \} } |\frac{\partial u_n}{\partial x_i} |^{p_i^-}dx \\ & \leq \sum _{i=1}^N\int _{\{ |u_n|\leq k \}}|\frac{\partial u_n}{\partial x_i} |^{p_i(x)}dx + N.\operatorname{meas}(\Omega)\\ & \leq \frac{C}{C_3}k + N.\operatorname{meas}(\Omega) \ \text{due to relation \eqref{eq3.10}} \\ & \leq C_5(1+k) \end{align*} with $ C_5 = \max \big\{ \frac{C}{C_3}; N.\operatorname{meas}(\Omega) \big\}$. \end{proof} We also have the following lemma (see \cite{Kone-Sado}, \cite{Kone-Ouaro}). \begin{lemma} \label{lem3.3} For any $k > 0$, there exists some positive constants $C_6$ and $C_7 $ such that: \begin{itemize} \item[ (i)] $\|u_n\|_{\mathcal{M}^{q^*}(\Omega)}\leq C_6$; \item[(ii)] $ \| \frac{\partial u_n}{\partial x_i} \| _{\mathcal{M}^{p_i^-\frac{q}{\overline{p}}}(\Omega)} \leq C_7, \ \ \forall i = 1,\dots ,N.$ \end{itemize} \end{lemma} \noindent\textbf{Step 3: Convergence.} According to \cite{Zongo Neumann} (see also \cite{Kone-Ouaro}), we have the following lemma. \begin{lemma} \label{lem3.4} For $i = 1,\dots ,N$, as $n\to +\infty$, we have \begin{equation} a_i\Big ( x, \frac{\partial u_n}{\partial x_i} \Big ) \to a_i \Big ( x, \frac{\partial u}{\partial x_i} \Big ) \quad \text{in }L^1(\Omega), \; \text{a.e. } x \in \Omega . \end{equation} \end{lemma} \begin{proposition} \label{prop3.1} Assume \eqref{eq1}-\eqref{eq6} hold. If $u_n \in E$ is a weak solution of \eqref{approx-problem} then, the sequence $(u_n)_{n\in \mathbb{N}^*}$ is Cauchy in measure. In particular, there exists a measurable function $u$ and a sub-sequence still denoted by $u_n$ such that $u_n \to u$ in measure. \end{proposition} \begin{proposition} \label{prop3.2} Assume \eqref{eq1}-\eqref{eq6} hold. If $u_n \in E$ is a weak solution of \eqref{approx-problem}, then \begin{itemize} \item[(i)] for $i = 1,\dots ,N$, $\frac{\partial u_n}{\partial x_i}$ converges in measure to the weak partial gradient of $u$; \item[(ii)] for $i = 1,\dots ,N$ and $k > 0$, $a_i\big( x, \frac{\partial }{\partial x_i}T_k(u_n) \big)$ converges to $a_i\big( x, \frac{\partial }{\partial x_i}T_k(u) \big)$ in $L^1(\Omega)$ strongly and in $L^{p'_i(\cdot)}(\Omega)$ weakly. \end{itemize} \end{proposition} We can now pass to the limit in \eqref{eq3.4}. Let $v\in W_0^{1,\overrightarrow{p}(\cdot)}(\Omega)\cap L^\infty (\Omega)$ and $k > 0$; we choose $T_k(u_n-v)$ as test function in \eqref{approximated solution} to obtain \begin{equation}\label{eq3.12} \begin{aligned} & \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u_n}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u_n-v)dx +\int _\Omega b(u_n)T_k(u_n - v)dx \\ &= \int _\Omega f_nT_k(u_n - v)dx +\int _\Omega F\nabla T_k(u_n - v)dx. \end{aligned} \end{equation} For the first term of the right-hand side of \eqref{eq3.12}, we have \begin{equation}\label{eq3.13} \int _\Omega f_n(x)T_k(u_n - v)dx \to \int _\Omega f(x)T_k(u - v)dx , \end{equation} since $f_n$ converges strongly to $f$ in $L^1(\Omega)$ and $T_k(u_n - v)$ converges weakly-$^*$ to $T_k(u - v)$ in $L^\infty (\Omega)$ and a.e. in $\Omega$. For the second term of the right-hand side of \eqref{eq3.12}, we have \begin{equation}\label{eq3.14} \int _\Omega F\nabla T_k(u_n - v)dx\to \int _\Omega F\nabla T_k(u - v)dx, \end{equation} since $\nabla T_k(u_n - v) \rightharpoonup \nabla T_k(u - v)$ in $(L^{p_m(\cdot)}(\Omega))^N$ and $F\in (L^{p'_m(\cdot)}(\Omega))^N $. For the first term of \eqref{eq3.12}, we have (see \cite{Zongo Neumann}): \begin{equation}\label{eq3.15} \begin{aligned} &\lim _{n\to +\infty}\inf \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u_n}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u_n-v)dx \\ &\geq \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - v)dx. \end{aligned} \end{equation} For the second term of \eqref{eq3.12}, we have \[ \int _\Omega b(u_n)T_k(u_n - v)dx = \int _\Omega (b(u_n) - b(v))T_k(u_n - v)dx + \int _\Omega b(v)T_k(u_n - v)dx. \] The quantity $(b(u_n) - b(v))T_k(u_n - v)$ is nonnegative and since for all $s\in \mathbb{R}$, $s\mapsto b(s)$ is continuous, we obtain \[ (b(u_n) - b(v))T_k(u_n - v) \to (b(u) - b(v))T_k(u - v) \quad\text{a.e. } \text{in }\Omega. \] Then, it follows by Fatou's Lemma that \begin{equation}\label{limit 1} \lim _{n\to +\infty}\inf \int _\Omega (b(u_n) - b(v))T_k(u_n - v)dx \geq \int _\Omega (b(u) - b(v))T_k(u - v)dx. \end{equation} We have $b(v)\in L^1(\Omega)$. Since $T_k(u_n - v)$ converges weakly -* to $T_k(u - v)\in L^\infty (\Omega)$ and $b(v) \in L^1 (\Omega)$, it follows that \begin{equation}\label{limit 2} \lim _{n\to +\infty}\int _\Omega b(v)T_k(u_n - v)dx = \int _\Omega b(v)T_k(u_n - v)dx. \end{equation} From \eqref{eq3.13}, \eqref{eq3.14}, \eqref{eq3.15}, \eqref{limit 1} and \eqref{limit 2}, we pass to the limit in \eqref{eq3.12} to obtain \begin{align*} &\sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u-v)dx +\int _\Omega b(u)T_k(u-v)dx\\ &\leq \int _\Omega fT_k(u-v)dx + \int _\Omega F\nabla T_k(u - v)dx. \end{align*} Then $u$ is an entropy solution of \eqref{e1.1}. \end{proof} \begin{theorem} \label{thm3.2} Assume \eqref{eq1}-\eqref{eq6} hold and let $u$ be an entropy solution of \eqref{e1.1}. Then, $u$ is unique. \end{theorem} The proof of the above theorem is done in two steps. \smallskip \noindent\textbf{Step 1: A priori estimates.} \begin{lemma} \label{lem3.5} Assume \eqref{eq1}-\eqref{eq6} hold. Let $u$ be an entropy solution of \eqref{e1.1}. Then \begin{equation}\label{eq3.25} \sum _{i=1}^N\int _{\{|u|\leq k\}}|\frac{\partial u}{\partial x_i} |^{p_i^-}dx \leq \frac{k}{C_3}|\mu|(\Omega), \quad \forall k > 0 \end{equation} and there exists a positive constant $C_8$ such that \begin{equation}\label{eq3.26} \|b(u)\|_1 \leq C_8 \operatorname{meas}(\Omega) + |\mu|(\Omega). \end{equation} \end{lemma} \begin{proof} Let us take $v = 0$ in the entropy inequality \eqref{solution}. \noindent$\bullet$ By the fact that $ \int _\Omega b(u)T_k(u)dx \geq 0$ and using relations \eqref{eq3} and \eqref{eq4}, we deduce \eqref{eq3.25}. \noindent$\bullet$ As \[ \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big )\frac{\partial }{\partial x_i}T_k(u)dx = \sum ^N_{i=1}\int _{\{|u|\leq k \}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial u}{\partial x_i}dx \geq 0, \] relation \eqref{solution} gives \begin{equation}\label{eq3.27} \int _\Omega b(u)T_k(u)dx \leq \int _\Omega fT_k(u)dx + \int _\Omega F\nabla T_k(u)dx. \end{equation} By \eqref{eq3.27}, we deduce \[ \int _{\{ |u|\leq k \}} b(u)T_k(u)dx + \int _{\{ |u|>k \}} b(u)T_k(u)dx \leq k|\mu|(\Omega) \] or \[ \int _{\{ |u|> k \}} b(u)T_k(u)dx = -k\int _{\{ u < -k \}} b(u)dx + k\int _{\{ u> k \}} b(u)dx \leq k|\mu |(\Omega) . \] Therefore, \[ \int _{\{ |u| > k \}} |b(u)|dx \leq |\mu |(\Omega) . \] So, we obtain \begin{align*} \int _\Omega |b(u)|dx &= \int _{\{ |u| > k \}} |b(u)|dx + \int _{\{ |u| \leq k \}} |b(u)|dx \\ & \leq \int _{\{ |u| \leq k \}} |b(u)|dx + |\mu|(\Omega). \end{align*} Since $b$ is non-decreasing, we have \[ |u|\leq k \Leftrightarrow b(-k)\leq b(u)\leq b(k) \Rightarrow |b(u)|\leq \max \{ b(k), |b(-k)| \}. \] Then, we have \[ \int _{\{ |u| \leq k \}} |b(u)|dx \leq \int _\Omega \max \{ b(k), |b(-k)| \}dx = \max \{ b(k), |b(-k)| \}\operatorname{meas}(\Omega). \] Consequently, there exists a constant $C_8 = \max \{ b(k), |b(-k)| \}$ such that \[ \|b(u)\|_1 \leq C_8\operatorname{meas}(\Omega) + |\mu|(\Omega). \] \end{proof} \begin{lemma} \label{lem3.6} Assume \eqref{eq1}-\eqref{eq6} hold true and $\mu \in \mathcal{M}_b^{p(\cdot)}(\Omega)$. If $u$ is an entropy solution of \eqref{e1.1}, then there exists a constant $D$ which depends on $\mu$ and $\Omega$ such that \begin{equation}\label{eq3.28} \operatorname{meas}\{ |u| > k \}\leq \frac{D}{\min (b(k), |b(-k)|)}, \forall k > 0 \end{equation} and a constant $D'$ which depends on $\mu$ and $\Omega$ such that \begin{equation}\label{eq3.29} \operatorname{meas} \big \{ | \frac{\partial u}{\partial x_i} | > k \big \} \leq \frac{D'}{k^{\frac{1}{(p_M)'}}}, \quad \forall k \geq 1. \end{equation} \end{lemma} \begin{proof} Since $b$ is non-decreasing, we have \[ \forall k > 0, |u| > k \Rightarrow |b(u)|\geq \min (b(k), |b(-k)|). \] For any $k > 0$, the relation \eqref{eq3.26} and the fact that $|b(u)|\geq \min (b(k), |b(-k)|)$ give \[ \int _{ \{|u| > k\}} \min (b(k), |b(-k)|)dx \leq \int _{\{|u| > k\}}|b(u)|dx \leq C_8\operatorname{meas}(\Omega) + |\mu|(\Omega). \] Therefore, \[ \min (b(k), |b(-k)|)\operatorname{meas}({|u| > k}) \leq C_8\operatorname{meas}(\Omega) + |\mu|(\Omega) = D ; \] that is \[ \operatorname{meas}({|u| > k}) \leq \frac{D}{\min (b(k), |b(-k)|)} . \] For the proof of \eqref{eq3.29}, we refer to \cite{Bonzi Ouaro}. \end{proof} We have the following two lemmas whose proofs can be found in \cite{Ibrango}. \begin{lemma} \label{lem3.7} Assume \eqref{eq1}-\eqref{eq6} hold, and let $f\in L^1(\Omega)$. If $u$ is an entropy solution of \eqref{e1.1}, then \[ \lim _{h\to +\infty} \int _\Omega |f|\chi _{ \{|u|>h-t\}}dx = 0, \] where $h > 0$ and $t > 0$. \end{lemma} \begin{lemma} \label{lem3.8} Assume \eqref{eq1}-\eqref{eq6} hold, and let $f \in L^1(\Omega)$. If $u$ is an entropy solution of \eqref{e1.1}, then there exists a positive constant $K$ such that \begin{equation} \rho _{p'_i(\cdot)} \Big ( | \frac{\partial u}{\partial x_i} |^{p_i(x)-1} \chi _{F_{h,k}} \Big )\leq K, \quad\text{for } i = 1,\dots ,N, \end{equation} where $F_{h,k} = \{ h <|u| \leq h + k \}$, $h > 0$, $k > 0$. \end{lemma} \noindent\textbf{Step 2: Uniqueness of the entropy solution.} Let $h > 0$ and $u, v$ be two entropy solutions of \eqref{e1.1}. We write the entropy inequality corresponding to the solution $u$, with $T_h(v)$ as test function, and to the solution $v$, with $T_h(u)$ as test function. We obtain \begin{equation}\label{eq3.32} \begin{aligned} &\sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx +\int _\Omega b(u)T_k(u - T_h(v) )dx\\ &\leq \int _\Omega fT_k(u - T_h(v))dx + \int _\Omega F\nabla T_k(u - T_h(v))dx \end{aligned} \end{equation} and \begin{equation}\label{eq3.33} \begin{aligned} &\sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial v}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(v - T_h(u))dx +\int _\Omega b(v)T_k(v - T_h(u) )dx\\ &\leq \int _\Omega fT_k(v - T_h(u))dx + \int _\Omega F\nabla T_k(v - T_h(u))dx. \end{aligned} \end{equation} Adding \eqref{eq3.32} and \eqref{eq3.33}, we obtain \begin{equation}\label{eq3.34} \begin{aligned} &\Big [ \sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big )\frac{\partial }{\partial x_i}T_k(u - T_h(v))dx - \int _\Omega F.\nabla T_k(u - T_h(v))dx\Big ] \\ &+\Big [ \sum ^N_{i=1} \int _\Omega a_i \Big (x,\frac{\partial v}{\partial x_i} \Big )\frac{\partial }{\partial x_i}T_k(v - T_h(u))dx - \int _\Omega F.\nabla T_k(v - T_h(u))dx\Big ] \\ &+ \int _\Omega b(u)T_k(u - T_h(v) )dx + \int _\Omega b(v)T_k(v - T_h(u) )dx \\ &\leq \int _\Omega f(x)[T_k(u - T_h(v)) + T_k(v - T_h(u)) ]dx. \end{aligned} \end{equation} Let us define \begin{gather*} E_1 = \{ |u - v|\leq k; |v| \leq h\}, \quad E_2 = E_1\cap \{|u| \leq h\}, \quad E_3 = E_1\cap \{ |u| > h \}, \\ E'_1 = \{ |v - u|\leq k; |u| \leq h\}, \quad E'_3 = E'_1\cap \{ |v| > h \} . \end{gather*} \smallskip \noindent\textbf{Assertion 1.} $E_3 \subset F_{h,k}$ and \[ B = \{ |u - h\operatorname{sign}_0 (v)|\leq k, |v| > h \} \subset F_{h - k, \ 2k}. \] Indeed, We decompose $E_3$ as $E_3 = E_3^+ \cup E_3^-$ where $E_3^+ = \{ |u-v|\leq k, |v|\leq h, u > h \}$ and $E_3^- = \{|u-v|\leq k, |v|\leq h, u < - h\}$. In $E_3^+$, we have $-h \leq v \leq h$ and $-k \leq u - v \leq k$ so that $v - k \leq u \leq v + k \leq h + k$. Since $u > h$ and $v+k\leq h+k$, we obtain $h \leq u \leq h + k$. Hence, $E_3^+\subset F_{h,k}$. In $E_3^-$, we have $-h \leq v \leq h$ and $-k \leq u - v \leq k$ so that $v - k \leq u \leq v + k \leq h + k$; since $u < -h$ and $- k - h \leq v - k \leq h - k$, we obtain $- h - k \leq u \leq - h$ so that $h \leq |u| \leq h + k$.\\ Hence, $E_3^-\subset F_{h,k}$. We split $B$ as $B = B^+\cup B^- = \{ |u-h|\leq k, v> h \} \cup \{ |u+h|\leq k, v < -h \}$. In $B^+$ ($B^-$ can be treated in the same way), we have $-k \leq u - h \leq k$ and $v > h$ so that $h-k \leq u \leq h+k$. Hence $B^+ \subset F_{h-k, 2k}$. \smallskip \noindent\textbf{Assertion 2.} On $E_3$ (and on $B$) we have according to H\"older inequality \begin{equation}\label{eq3.35} \int _{E_3} F.\nabla udx \leq \Big (\int _{E_3} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _{E_3} |\nabla u|^{p_m^-}dx \Big )^\frac{1}{(p_m)^-}, \end{equation} with \[ \lim _{h\to +\infty} \Big (\int _{E_3} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _{E_3} |\nabla u|^{(p_m^-)}dx \Big )^\frac{1}{(p_m^-)} = 0, \] where $\frac{1}{(p_m^-)} + \frac{1}{(p'_m)^-} = 1$. Indeed, \[ \lim _{n\to +\infty}\Big ( \int _{E_3} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} = 0, \] as $|F|^{(p'_m)^-}\chi _{E_3}$ belongs to $L^1(\Omega)$ and as $E_3\subset F_{h,k}$, then $|F|^{(p'_m)^-}\chi _{E_3}$ converges to zero as $h\to +\infty$. Then, by Lebesgue dominated convergence theorem, \[ \lim _{h\to +\infty}\int _{E_3} |F|^{(p'_m)^-}dx = 0. \] Now, it remains to prove that $ \int _{E_3} |\nabla u|^{(p_m^-)}dx $ is bounded with respect to $h$. We use the notation. \[ \mathcal{I}_1 = \{ i\in \{1,\dots ,N\}: |\frac{\partial u}{\partial x_i} | \leq 1\}, \quad \mathcal{I}_2 = \{ i\in \{1,\dots ,N\}: |\frac{\partial u}{\partial x_i} | > 1 \}. \] Then we have \begin{align*} \sum ^N_{i=1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx & = \sum _{i\in \mathcal{I}_1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx + \sum _{i\in \mathcal{I}_2}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx\\ & \geq \sum _{i\in \mathcal{I}_2}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx\\ & \geq \sum _{i\in \mathcal{I}_2}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_m^-}dx\\ & \geq \sum ^N_{i=1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_m^-}dx - \sum _{i\in \mathcal{I}_1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_m^-}dx\\ & \geq \sum ^N_{i=1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_m^-}dx - N\operatorname{meas}(\Omega)\\ & \geq \sum ^N_{i=1}\| \frac{\partial u}{\partial x_i} \| _{L^{p_m^-}(F_{h, k})} ^{p_m^-}- N\operatorname{meas}(\Omega)\\ & \geq C_9 \|\nabla u\| _{(L^{p_m^-}(F_{h, k}))^N} ^{p_m^-} - N\operatorname{meas}(\Omega), \end{align*} where we used Poincar\'e inequality. We deduce that \begin{equation}\label{eqnew1} \sum ^N_{i=1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx \geq C_9 \int _ {F_{h, k}} |\nabla u|^{p_m^-} - N\operatorname{meas}(\Omega). \end{equation} Choosing $T_h(u)$ as test fonction in \eqref{solution}, we obtain \begin{equation}\label{num1} \begin{aligned} &\sum ^N_{i=1}\int _\Omega a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(u))dx + \int _\Omega b(u)T_k(u - T_h(u))dx \\ &\leq \int _\Omega fT_k(u - T_h(u))dx + \int _\Omega F.\nabla T_k(u - T_h(u))dx. \end{aligned} \end{equation} According to the fact that $\nabla T_k(u - T_h(u)) = \nabla u$ on $\{h \leq |u| \leq h + k\}$, and zero elsewhere, and $ \int _\Omega b(u)T_k(u - T_h(u))dx \geq 0$, we deduce from \eqref{num1} that \begin{equation}\label{num2} \begin{aligned} &\sum ^N_{i=1}\int _{F_{h, k}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial u}{\partial x_i}dx\\ &\leq k \int _{\{ |u| \geq h\}}|f|dx + \int _{F_{h, k}}|\Big (\frac{2}{C_3C_9 p^-_m}\Big )^{\frac{1}{p_m^-}}F| |\Big (\frac{C_3C_9 p_m^-}{2}\Big )^{\frac{1}{p_m^-}}\nabla u |dx. \end{aligned} \end{equation} Using \eqref{eq4} (in the left hand side of \eqref{num2}), Young inequality (in the right hand side of \eqref{num2}) and setting \[ C_{10} = \Big (\frac{2}{C_3C_9 p^-_m}\Big )^{\frac{(p'_m)^-}{(p_m^-)}} \frac{p_m^- - 1}{p_m^-} , \] we obtain \begin{equation}\label{eqnew2} \begin{aligned} &C_3\sum ^N_{i=1}\int _{F_{h, k}} | \frac{\partial u}{\partial x_i} |^{p_i(x)}dx \\ &\leq k \int _{\{ |u| \geq h\}}|f|dx + C_{10}\int _{F_{h, k}} |F|^{(p'_m)^-}dx + \frac{C_3C_9}{2}\int _{F_{h, k}} |\nabla u|^{p_m^-} dx. \end{aligned} \end{equation} From \eqref{eqnew1} and \eqref{eqnew2}, we obtain \begin{align*} &C_3C_9 \int _ {F_{h, k}} |\nabla u|^{p_m^-}dx\\ &\leq k \int _{\{ |u| \geq h\}}|f|dx + C_{10}\int _{F_{h, k}} |F|^{(p'_m)^-}dx + \frac{C_3C_9}{2}\int _{F_{h, k}} |\nabla u|^{p_m^-} dx + N\operatorname{meas}(\Omega). \end{align*} Therefore, \begin{equation}\label{eqnew3} \begin{aligned} &\frac{C_3C_9}{2} \int _ {F_{h, k}} |\nabla u|^{p_m^-}dx\\ &\leq k \int _{\{ |u| \geq h\}}|f|dx + C_{10}\int _{F_{h, k}} |F|^{(p'_m)^-}dx + N\operatorname{meas}(\Omega). \end{aligned} \end{equation} Since $E_3\subset F_{h,k}$, we deduce from \eqref{eqnew3} that $ \int _ {E_3} |\nabla u|^{p_m^-}dx$ is bounded. Since $B\subset F_{h - k,2k}$, reasoning as before, we obtain \[ \int _B F.\nabla udx \leq \Big (\int _B |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _B |\nabla u|^{p_m^-}dx \Big )^\frac{1}{p_m^-}, \] with \[ \lim _{h\to +\infty} \Big (\int _{B} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _{B} |\nabla u|^{(p_m^-)}dx \Big )^\frac{1}{(p_m^-)} = 0. \] We have \begin{align*} &\sum ^N_{i=1}\int _{\{|u-T_h(v)|\leq k\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx \\ &- \int _{\{|u-T_h(v)|\leq k\}} F \nabla T_k(u - T_h(v))dx\\ & = \sum ^N_{i=1}\int _{\{|u-T_h(v)|\leq k\}\cap \{|v|\leq h\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx \\ &\quad + \sum ^N_{i=1}\int _{\{|u-T_h(v)|\leq k\}\cap \{|v|> h\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx \\ &\quad - \int _{\{ |u - T_k(v)|\leq k\}\cap \{ |v|\leq h \}} F \nabla T_k(u - T_h(v))dx \\ &\quad - \int _{\{ |u - T_k(v)|\leq k \}\cap \{ |v| > h \}} F \nabla T_k(u - T_h(v))dx \\ & = \sum ^N_{i=1}\int _{\{|u - v|\leq k\}\cap \{|v|\leq h\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big )\frac{\partial}{\partial x_i}(u - v)dx\\ &\quad + \sum ^N_{i=1}\int _{\{|u - h\operatorname{sign}_0 (v)|\leq k\} \cap \{|v|\leq h\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial u}{\partial x_i}dx \\ &\quad - \int _{\{ |u - v|\leq k\}\cap \{ |v|\leq h \}} F \nabla (u - v)dx \\ &\quad - \int _{\{ |u - h\operatorname{sign}_0(v)|\leq k\}\cap \{ |v| > h \}} F \nabla T_k(u - h\operatorname{sign}_0(v))dx \\ & \geq \sum ^N_{i=1}\int _{E_1} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v)dx \\ &\quad - \int _{E_1} F \nabla (u - v)dx - \int _{\{ |u - h\operatorname{sign}_0(v)| \leq k\}\cap \{ |v| > h \}} F \nabla u\,dx \\ & = \sum ^N_{i=1}\int _{E_2} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v)dx + \sum ^N_{i=1}\int _{E_3} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big )\frac{\partial }{\partial x_i} (u - v)dx\\ &\quad - \int _{E_2} F \nabla (u - v)dx - \int _{E_3} F \nabla (u - v)dx - \int _{\{ |u - h\operatorname{sign}_0(v)|\leq k\}\cap \{ |v| > h \}} F \nabla u\,dx. \end{align*} Then, we obtain \begin{align*} &\sum ^N_{i=1}\int _{\{|u-T_h(v)|\leq k\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx \\ &- \int _{\{|u-T_k(v)|\leq k\}} F.\nabla T_k(u - T_h(v))dx \\ &\geq \sum ^N_{i=1}\int _{E_2} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v)dx + \sum ^N_{i=1}\int _{E_3} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v)dx \\ &\quad - \int _{E_2} F \nabla (u - v)dx + \int _{E_3} F \nabla vdx - \int _{E_3} F \nabla udx - \int _{B} F \nabla udx. \end{align*} We deduce from \eqref{eq3.35} that \begin{align} &\sum ^N_{i=1}\int _{\{|u-T_h(v)|\leq k\}} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial }{\partial x_i}T_k(u - T_h(v))dx \nonumber \\ &\quad - \int _{\{|u-T_k(v)|\leq k\}} F\nabla T_k(u - T_h(v))dx \nonumber\\ &\geq \sum ^N_{i=1}\int _{E_2} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v)dx \nonumber\\ &\quad -\Big [ \sum ^N_{i=1}\int _{E_3} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big )\frac{\partial}{\partial x_i}vdx - \int _{E_3} F \nabla vdx\Big ] \nonumber\\ &\quad - \int _{E_2} F \nabla (u - v)dx - \Big (\int _{E_3} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _{E_3} |\nabla u|^{(p_m^-)}dx \Big )^\frac{1}{(p_m^-)} \nonumber\\ &\quad - \Big ( \int _{B} |F|^{(p'_m)^-}dx \Big )^\frac{1}{(p'_m)^-} \Big (\int _{B} |\nabla u|^{(p_m^-)}dx \Big )^\frac{1}{p_m^-}. \label{eq3.37} \end{align} According to \eqref{eq2} and the H\"older type inequality, we have \begin{align*} &\big| \int _{E_3} \Big [\sum ^N_{i=1} a_i \Big (x,\frac{\partial u}{\partial x_i} \Big ) \frac{\partial}{\partial x_i}(u - v) - F.\nabla v \Big ]dx \big| \\ & \leq C_1 \sum ^N_{i=1}\int _{E_3} \Big ( j_i(x) + | \frac{\partial u}{\partial x_i} |^{p_i(x)-1} \Big )| \frac{\partial v}{\partial x_i} |dx + \int _{E_3} |F \nabla v|dx \\ & \leq C_1 \sum ^N_{i=1}\int _{E_3} \Big ( |j_i(x)| + \big|\, |\frac{\partial u}{\partial x_i} |^{p_i(x)-1} \big|_{p'_i(\cdot), \{h<|u|