\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 12, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/12\hfil Existence and uniqueness] {Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents} \author[B. K. Bonzi, I. Nyanquini, S. Ouaro \hfil EJDE-2012/12\hfilneg] {Bernard K. Bonzi, Ismael Nyanquini, Stanislas Ouaro} % in alphabetical order \address{Bernard K. Bonzi \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ UFR. Sciences Exactes et Appliqu\'ees, Universit\'e de Ouagadougou \\ 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso} \email{bonzib@univ-ouaga.bf} \address{Ismael Nyanquini \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ Institut des Sciences Exactes et Appliqu\'ees, Universit\'e Polytechnique de Bobo Dioulasso \\ 01 BP 1091 Bobo-Dioulasso 01 \\ Bobo Dioulasso, Burkina Faso} \email{nyanquis@yahoo.fr} \address{Stanislas Ouaro \newline Laboratoire d'Analyse Math\'ematique des Equations (LAME)\\ 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 13, 2011. Published January 17, 2012.} \subjclass[2000]{35J20, 35J25, 35D30, 35B38, 35J60} \keywords{Elliptic equation; weak solution; entropy solution; \hfill\break\indent Leray-Lions operator; variable exponent} \begin{abstract} In this article we study the nonlinear homogeneous Neumann boundary-value problem \begin{gather*} b(u)-\operatorname{div} a(x,\nabla u)=f\quad \text{in } \Omega\\ a(x,\nabla u).\eta=0 \quad\text{on }\partial \Omega, \end{gather*} where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 3$ and $\eta$ the outer unit normal vector on $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$ and the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponents. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The paper is motivated by phenomena which are described by a homogeneous Neumann boundary value problem of the type \begin{equation} \label{e1.1} \begin{gathered} b(u)-\operatorname{div} a(x,\nabla u)=f \quad\text{in } \Omega,\\ a(x,\nabla u).\eta=0 \quad\text{on }\partial \Omega, \end{gathered} \end{equation} where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 3$ and $\eta$ the outer unit normal vector on $\partial\Omega$. The study of problems involving variable exponents has received considerable attention in recent years (see \cite{b1}, \cite{b3}-\cite{g1}, \cite{k1}-\cite{m2}, \cite{o1}-\cite{s1}, \cite{w1}-\cite{z1}) due to the fact that they can model various phenomena which arise in the study of elastic mechanics (see \cite{a4}), electrorheological fluids (see \cite{d2,m1,r1,r2}) or image restauration (see \cite{c1}). When the boundary value condition is a Neumann boundary condition in the context of variable exponent, we must work in general with the space $W^{1,p(\cdot)}(\Omega)$ instead of the common space $W^{1,p(\cdot)}_0(\Omega)$. The main difficulty which appears in this case for the existence and also the uniqueness of solutions is that the famous Poincar\'e inequality does not apply (see \cite{b4}). The same can be said for the Poincar\'e-Wirtinger inequality which does not apply for general data $f$ considered in this work (see \cite{o2}). Recently, Ouaro and Soma \cite{o2} studied the problem \begin{equation} \label{e1.2} \begin{gathered} -\operatorname{div} a(x,\nabla u) + |u|^{p(x)-2}u = f\quad \text{in } \Omega,\\ \frac{\partial u}{ \partial \nu} = 0 \quad \text{on }\partial \Omega, \end{gathered} \end{equation} under the assumption \begin{equation} \label{e1.3} p(\cdot):\Omega\to\mathbb{R} \text{ is a measurable function and } 1< p_{-}\leq p_{+} < +\infty, \end{equation} where $p_{-}:=\operatorname{ess\,inf}_{x\in \Omega}p(x)$ and $p_{+}:=\operatorname{ess\,sup}_{x\in\Omega}p(x)$. For the vector fields $a(\cdot,\cdot)$ in \cite{o2}, the authors assumed that $a(x,\xi):\Omega\times\mathbb{R}^{N}\to\mathbb{R}^{N}$ is Carath\'eodory and is the continuous derivative with respect to $\xi$ of the mapping $A: \Omega\times\mathbb{R}^{N}\to\mathbb{R}$, $A=A(x,\xi)$; i.e., $a(x,\xi)=\nabla_{\xi}A(x,\xi)$ such that: \begin{itemize} \item for almost every $x\in\Omega$, \begin{equation} \label{e1.4} A(x,0)=0; \end{equation} \item there exists a positive constant $C_1$ such that \begin{equation} \label{e1.5} |a(x,\xi)|\leq C_1(j(x) + |\xi|^{p(x)-1}) \end{equation} for almost every $x\in\Omega$ and for every $\xi\in \mathbb{R}^{N}$ where $j$ is a nonnegative function in $L^{p'(\cdot)}(\Omega)$, with $ 1/p(x) + 1/p'(x) = 1$; \item the following inequality hold for almost every $x \in \Omega $ and for every $\xi, \eta \in \mathbb{R}^{N}$ with $\xi \neq \eta$, \end{itemize} \begin{equation} \label{e1.6} ( a(x,\xi) - a(x,\eta)).(\xi - \eta) > 0; \end{equation} \begin{itemize} \item for almost every $x\in \Omega$ and for every $\xi\in\mathbb{R}^{N}$, \begin{equation} \label{e1.7} |\xi|^{p(x)}\leq a(x,\xi).\xi\leq p(x)A(x,\xi) \end{equation} \end{itemize} Under assumptions \eqref{e1.3}-\eqref{e1.7}, Ouaro and Soma \cite{o2} proved the existence and uniqueness of entropy solutions to \eqref{e1.2} for $L^{1}-$data $f$. The assumption on the function $A$ and the use of the quantity $|u|^{p(x)-2}u$ allowed them in particular to use a minimization method for the proof of the existence of a weak solution for \eqref{e1.2} when the right-hand side is in $L^{\infty}(\Omega)$ (see \cite[Theorem 3.1]{o2}). Note also that the uniqueness of weak and entropy solutions $u$ of \cite[(1.2)]{o2} is due to the fact that $s\mapsto|s|^{p(x)-2}s$ is increasing. In this article we improve the result in \cite{o2}. We make restrictive assumptions on the data $a$ and $b$. For this reason, we can not use the minimization methods used in \cite{o2} to get our existence result of weak solutions. We use an auxiliary result due to Le (see \cite[Theorem 3.1]{l1}). Indeed, Le \cite{l1} proved in particular some existence results of weak solutions for the Neumann and Robin boundary value problem \begin{gather*} -\operatorname{div} a(x,\nabla u)+f(x,u)=0 \quad\text{in } \Omega,\\ a(x,\nabla u).\eta=-g(x,u)\quad\text{on }\partial \Omega, \end{gather*} where $a:\Omega\times\mathbb{R}^{N}\to \mathbb{R}$ is a Carath\'eodory function satisfying the growth condition \[ |a(x,\xi)|\leq a_1(x)+b_1|\xi|^{p(x)-1},\quad \text{for a. e. $x\in \Omega$ and all }\xi\in\mathbb{R}^{N}, \] with $p\in C_{+}(\overline{\Omega}) =\{p\in C(\overline{\Omega})\text{ such that }p(x)>1 \text{ for }x\in \overline{\Omega}\}$, $a_1\in L^{p'(\cdot)}(\Omega)$, $p'(\cdot)$ is the H\"older conjugate of $p(\cdot)$ and $b_1>1$. Moreover, $a$ is monotone; i.e., \[ (a(x,\xi)-a(x,\xi')).(\xi-\xi')\geq 0,\quad \text{ for a. e. $x\in\Omega$ and all }\xi,\xi'\in \mathbb{R}^{N}, \] and coercive in the following sense: there exist $a_2\in L^{1}(\Omega)$ and $b_2>0$ such that \[ a(x,\xi).\xi\geq b_2|\xi|^{p(x)}-a_2(x), \quad\text{for a. e. $x\in\Omega$ and all }\xi\in\mathbb{R}^{N}. \] $f:\Omega\times \mathbb{R}\to\mathbb{R}$ and $g:\partial\Omega\to \mathbb{R}$ are Carath\'eodory functions such that \[ |f(x,u)|\leq a_3(x),\quad |g(\xi,v)|\leq \widetilde{a}_3(\xi) \] for a. e. $x\in \Omega$, $\xi\in\partial\Omega$, where $a_3\in L^{q(\cdot)}(\Omega)$, $\widetilde{a}_3\in L^{\widetilde{q}}(\partial\Omega)$ with $q(x)< p^{*}(x)$, for all $x\in\overline{\Omega}$, $\widetilde{q}(x)<\widetilde{p}^{*}(x)$, for all $x\in \partial\Omega$, $q\in C_{+}(\overline{\Omega})$, $\widetilde{q}\in C_{+}(\partial\Omega)$. Here, $p^{*}$ is the Sobolev conjugate exponent of $p(x)$, \begin{gather*} p^{*}(x)=\begin{cases} \frac{Np(x)}{N-p(x)} &\text{if }N>p(x),\\ +\infty &\text{if }N\leq p(x); \end{cases} \\ \widetilde{p}^{*}(x)=\begin{cases} \frac{(N-1)p(x)}{N-p(x)}&\text{if }N>p(x),\\ +\infty &\text{if }N\leq p(x). \end{cases} \end{gather*} The proof of the existence results in \cite{l1} uses the sub and super solution methods. In this article, our assumptions are the following: \begin{equation} \label{e1.8} \text{$p(\cdot): \overline{\Omega} \to \mathbb{R}$ is a continuous function such that $10, \\ a(x,\xi).\xi\geq C_3|\xi|^{p(x)} \label{e1.12} \end{gather} for almost every $x \in \Omega$ and for every $\xi \in \mathbb{R}^N$. \end{itemize} We remark that \cite[Assumption 1.3]{o2} is more restrictive than \eqref{e1.8}. This is due to the use of the results in \cite{l1} to get the existence of a weak solution to the problem \eqref{e1.1}. The remaining part of the paper is the following: in section 2, we introduce some notations/functional spaces. In section 3, we prove the existence and uniqueness of a weak solution of \eqref{e1.1} when the right-hand side $f\in L^{\infty}(\Omega)$. Using the results of section 3, we study in section 4, the question of the existence and uniqueness of entropy solutions of \eqref{e1.1} for $f\in L^{1}(\Omega)$. \section{Assumptions and preliminaries} As the exponent $p(\cdot)$ appearing in \eqref{e1.10} and \eqref{e1.12} depends on the variable $x$, we must work with Lebesgue and Sobolev spaces with variable exponents. We define the Lebesgue space with variable exponent $L^{p(\cdot)}(\Omega)$ as the set of all measurable functions $u: \Omega \to \mathbb{R}$ for which the convex modular $$ \rho_{p(\cdot)}(u):= \int_{\Omega}|u|^{p(x)}dx $$ is finite. If the exponent is bounded; i.e., if $p_{+}< +\infty$, then the expression $$ |u|_{p(\cdot)}=\inf \{\lambda >0: \rho_{p(\cdot)}(u/ \lambda) \leq 1 \} $$ defines a norm in $L^{p(\cdot)}(\Omega)$, called the Luxembourg norm. The space $(L^{p(\cdot)}(\Omega),|.|_{p(\cdot)})$ is a separable Banach space. Moreover, if $11 \Rightarrow |u|^{p_{-}}_{p(\cdot)} \leq\rho_{p(\cdot)}(u)\leq |u|^{p_{+}}_{p(\cdot)}$; \item[(ii)] $|u|_{p(\cdot)}<1 \Rightarrow |u|^{p_{+}}_{p(\cdot)} \leq\rho_{p(\cdot)}(u)\leq |u|^{p_{-}}_{p(\cdot)}$; \item[(iii)] $|u|_{p(\cdot)}<1$ (respectively $=1;>1$) $\Leftrightarrow \rho_{p(\cdot)}(u)<1$ (respectively $=1;>1$); \item[(iv)] $|u_n|_{p(\cdot)}\to 0$ (respectively $ \to +\infty$) $\Leftrightarrow \rho_{p(\cdot)}(u_n) \to 0$ (respectively $\to +\infty$); \item[(v)] $\rho_{p(\cdot)}(u/|u|_{p(\cdot)})=1$ \end{itemize} \end{lemma} For a measurable function $u: \Omega \to \mathbb{R}$, we introduce the function $$ \rho_{1,p(\cdot)}(u)= \int_{\Omega}|u|^{p(x)}\,dx +\int_{\Omega}|\nabla u|^{p(x)}\,dx. $$ Then we have the following lemma (see \cite{w1,y1}). \begin{lemma} \label{lem2.2} If $u \in W^{1,p(\cdot)}(\Omega)$, then the following properties hold: \begin{itemize} \item[(i)] $\|u\|_{1,p(\cdot)}>1 \Rightarrow \|u\|^{p_{-}}_{1,p(\cdot)}\leq \rho_{1,p(\cdot)}(u) \leq \|u\|_{1,p(\cdot)}^{p_{+}}$; \item[(ii)] $\|u\|_{p(\cdot)}<1 \Rightarrow \|u\|_{1,p(\cdot)}^{p_{+}} \leq \rho_{1,p(\cdot)}(u) \leq \|u\|^{p_{-}}_{1,p(\cdot)}$; \item[(iii)] $\|u\|_{1,p(\cdot)}<1$ (respectively $=1;>1$) $\Leftrightarrow \rho_{1,p(\cdot)}(u)<1$ (respectively $=1;>1$); \end{itemize} \end{lemma} Given two bounded measurable functions $p(\cdot),q(\cdot): \Omega \to \mathbb{R}$, we write $$ q(\cdot)\ll p(\cdot) \quad \text{if } \operatorname{ess \,inf}_{x \in \Omega}(p(x)-q(x))>0. $$ For more details about Lebesgue and Sobolev spaces with variable exponent, we refer to \cite{d1,m3,n1,s1,t1,z1} and the references therein. \section{Existence and uniqueness of weak solutions} In this part, we study the existence and uniqueness of a weak solution of \eqref{e1.1} for the right-hand side $f \in L^{\infty}(\Omega)$. The concept of uniqueness is the same as in \cite{a2}. \begin{definition} \label{def3.1}\rm A weak solution of \eqref{e1.1} is a measurable function such that $$ u \in W^{1,p(\cdot)}(\Omega),\quad b(u) \in L^{\infty}(\Omega) $$ and \begin{equation} \label{e3.1} \int_{\Omega} a(x,\nabla u).\nabla \varphi \,dx+ \int_{\Omega} b(u) \varphi \,dx=\int_{\Omega} f \varphi \,dx, \; \quad \forall \varphi \in W^{1,p(\cdot)}(\Omega). \end{equation} \end{definition} The main result of this part is the following. \begin{theorem} \label{thm3.2} Assume that \eqref{e1.8}--\eqref{e1.12} hold true and $f \in L^{\infty}(\Omega)$. Then there exists a unique weak solution of \eqref{e1.1}. \end{theorem} \begin{proof} (Part 1: Existence). For $k>0$, we consider the following approximated problem. \begin{equation} \label{e3.2} \begin{gathered} T_k(b(u_k))-\operatorname{div} a(x,\nabla u_k)=f \quad\text{in } \Omega\\ a(x,\nabla u_k).\eta=0 \quad\text{on }\partial \Omega, \end{gathered} \end{equation} where $T_{k}(s) :=\max\{-k, \min\{k,s \}\}$ is the truncation of $T_{k}$, for any $k>0$. Note that as $T_k(b(u_k)) \in L^{\infty}(\Omega)$, by \cite[Theorem 3.1]{l1}, there exists $u_k \in W^{1,p(\cdot)}(\Omega)$ which is a weak solution of \eqref{e3.2}. We now show that $|b(u_k)| \leq \|f\|_{\infty}$ for all $k>0$. We recall that for any $\epsilon >0$, \begin{gather*} H_{\epsilon}(s)=\min(\frac{s^{+}}{\epsilon},1), \\ \operatorname{sign}_0^{+}(s)=\begin{cases} 1 &\text{if }s>0\\ 0 &\text{if }s\leq 0 \end{cases} \end{gather*} and 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 take $\varphi=H_{\epsilon}(u_k-M)$ as a test function in \eqref{e3.1} for the weak solution $u_{k}$ and $M>0$ a constant to be chosen later. We have \begin{equation} \label{e3.3} \int_{\Omega} a(x,\nabla u_k).\nabla H_{\epsilon}(u_k-M) \,dx + \int_{\Omega} T_k(b(u_k)) H_{\epsilon}(u_k-M) \,dx =\int_{\Omega} f H_{\epsilon}(u_k-M)dx. \end{equation} Let us denote $J= \int_{\Omega} a(x,\nabla u_k).\nabla H_{\epsilon}(u_k-M) \,dx$. We deduce that $$ J= \frac{1}{\epsilon} \int_{\{0< u_k-M< \epsilon\}} a(x,\nabla u_k).\nabla (u_k-M) \,dx \geq 0, $$ Then, according to \eqref{e3.3}, we obtain $$ \int_{\Omega} T_k(b(u_k)) H_{\epsilon}(u_k-M) \,dx \leq \int_{\Omega} f H_{\epsilon}(u_k-M) \,dx, $$ which is equivalent to saying \begin{equation} \label{e3.4} \int_{\Omega} (T_k(b(u_k))-T_k(b(M))) H_{\epsilon}(u_k-M) \,dx \leq \int_{\Omega} (f-T_k(b(M)) )H_{\epsilon}(u_k-M) \,dx. \end{equation} We now let $\epsilon$ approach $0$ in the above inequality, \begin{equation} \label{e3.5} \int_{\Omega} (T_k(b(u_k))-T_k(b(M)))^+ \,dx \leq \int_{\Omega} (f-T_k(b(M)) )\operatorname{sign}_0^{+}(u_k-M) \,dx. \end{equation} Choosing now $M=b_0^{-1}(\|f\|_{\infty})$ in \eqref{e3.5} (since $b$ is surjective) to obtain \begin{equation} \label{e.36} \int_{\Omega} (T_k(b(u_k))-T_k(\|f\|_{\infty}))^+ \,dx \leq \int_{\Omega} (f-T_k(\|f\|_{\infty}) )\operatorname{sign}_0^{+}(u_k-b_0^{-1}(\|f\|_{\infty})) \,dx. \end{equation} Hence for all $k>\|f\|_{\infty}$, we have $$ \int_{\Omega} (T_k(b(u_k))-T_k(\|f\|_{\infty}))^+ \,dx \leq \int_{\Omega} (f-\|f\|_{\infty} ) \operatorname{sign}_0^{+}(u_k-b_0^{-1}(\|f\|_{\infty})) \,dx \leq 0. $$ Then for all $k>\|f\|_{\infty}$, $(T_k(b(u_k))-\|f\|_{\infty})^+=0$ a.e. in $\Omega$ which is equivalent to saying \begin{equation} \label{e3.7} T_k(b(u_k))\leq \|f\|_{\infty}\quad \text{for all }k>\|f\|_{\infty}. \end{equation} It remains to prove that $T_k(b(u_k))\geq -\|f\|_{\infty}$ a.e. in $\Omega$ for all $k>\|f\|_{\infty}$. \end{proof} Let us remark that as $u_{k}$ is a weak solution of \eqref{e3.2}, then $(-u_k)$ is a weak solution to the following problem \begin{equation} \label{e3.8} \begin{gathered} T_k(\tilde{b}(u_k))-\operatorname{div} \tilde{a}(x,\nabla u_k) =\tilde{f} \quad\text{in } \Omega\\ \tilde{a}(x,\nabla u_k).\eta=0 \quad\text{on }\partial \Omega, \end{gathered} \end{equation} where $\tilde{a}(x,\xi)=-a(x,-\xi)$, $\tilde{b}(s)=-b(-s)$ and $\tilde{f}=-f$. According to \eqref{e3.7}, we deduce that \[ T_k(-b(u_k))\leq \|f\|_{\infty} \quad\text{a.e. in $\Omega$ for all } k>\|f\|_{\infty}. \] Therefore, we obtain \begin{equation} \label{e3.9} T_k(b(u_k))\geq -\|f\|_{\infty} \quad \forall k>\|f\|_{\infty}. \end{equation} It follows from \eqref{e3.7} and \eqref{e3.9} that for all $k>\|f\|_{\infty}$, $|T_k(b(u_k))|\leq \|f\|_{\infty}$ which implies \begin{equation} \label{e3.10} |b(u_k)|\leq \|f\|_{\infty} \quad \text{a.e. in } \Omega. \end{equation} We now fix $k=\|f\|_{\infty}+1$ in \eqref{e3.2} to end the proof of the existence result. {Part 2: Uniqueness.} Let $u_1$ and $u_2$ be two weak solutions of \eqref{e1.1}. Let us take $\varphi=T_k(u_1-u_2)$ as a test function in \eqref{e3.1} for $u_1$ and also for $u_2$, to get $$ \int_{\Omega} a(x,\nabla u_1).\nabla T_k(u_1-u_2) \,dx + \int_{\Omega} b(u_1)T_k(u_1-u_2) \,dx =\int_{\Omega} f T_k(u_1-u_2) \,dx, $$ and $$ \int_{\Omega} a(x,\nabla u_2).\nabla T_k(u_1-u_2) \,dx + \int_{\Omega} b(u_2)T_k(u_1-u_2) \,dx =\int_{\Omega} f T_k(u_1-u_2) \,dx. $$ Adding the two preceding relations, we obtain \begin{equation} \label{e3.11} \int_{\Omega} (a(x,\nabla u_1)-a(x,\nabla u_2)).\nabla T_k(u_1-u_2) \,dx+ \int_{\Omega}(b(u_1)- b(u_2))T_k(u_1-u_2) \,dx=0. \end{equation} From \eqref{e3.11} we deduce that \begin{gather} \label{e3.12} \int_{\Omega} (a(x,\nabla u_1)-a(x,\nabla u_2)).\nabla T_k(u_1-u_2) \,dx=0, \\ \label{e3.13} \int_{\Omega}(b(u_1)- b(u_2))T_k(u_1-u_2) \,dx=0. \end{gather} Thanks to \eqref{e3.12} and inequality \eqref{e1.11}, we obtain \begin{equation} \label{e3.14} u_1-u_2=c \quad\text{a.e. in }\Omega \end{equation} and the relation \eqref{e3.13} gives \[ \lim_{k\to 0} \int_{\Omega}(b(u_1)- b(u_2))\frac{1}{k}T_k(u_1-u_2) \,dx = \int_{\Omega}|b(u_1)- b(u_2)| \,dx=0. \] Finally, we obtain \begin{equation} \label{e3.15} \begin{gathered} u_1-u_2=c \quad \text{a.e. in }\Omega\\ \text{and } b(u_1)=b(u_2). \end{gathered} \end{equation} \section{Entropy solutions} In this section, we study the existence and uniqueness of entropy solutions to problem \eqref{e1.1} when the right-hand side $f \in L^{1}(\Omega)$. We first recall some notations. Set \[ \mathcal{T}^{1,p(\cdot)}(\Omega) = \{ u: \Omega \to \mathbb{R}, \text{ measurable such that $T_{k}(u) \in W^{1,p(\cdot)}(\Omega)$ for any } k >0 \}. \] As in \cite{b2} (see also \cite{a1}), we can prove the following result. \begin{proposition} \label{prop4.1} Let $u \in \mathcal{T}^{1,p(\cdot)}(\Omega)$. Then there exists a unique measurable function $ v: \Omega \to \mathbb{R}^{N}$ such that $ \nabla T_{k}(u) = v\chi_{\{|u| < k\}}$ for all $k>0$. The function $v$ is denoted by $\nabla u$. Moreover, if $u \in W^{1,p(\cdot)}(\Omega)$ then $v \in (L^{p(\cdot)}(\Omega))^{N}$ and $v=\nabla u$ in the usual sense. \end{proposition} We define $\mathcal{T}^{1,p(\cdot)}_{\mathcal{H}}(\Omega)$ as the set of functions $u \in \mathcal{T}^{1,p(\cdot)}(\Omega)$ such that there exists a sequence $(u_{n})_{n} \subset W^{1,p(\cdot)}(\Omega) $ satisfying the following conditions: \begin{itemize} \item[(C1)] $u_{n} \to u$ a.e. in $\Omega$. \item[(C2)] $\nabla T_{k}(u_{n}) \to \nabla T_{k}(u)$ in $L^{1}(\Omega)$ for any $k>0$. \end{itemize} The symbol $\mathcal{H}$ in the notation is related to the fact that we consider here Homogeneous Neumann Boundary condition. For the Nonhomogeneous Neumann Boundary condition, we need to add the definition of the set in the following boundary condition, to give meaning to the solution at the boundary. \begin{itemize} \item[(C3)] There exists a measurable function $v$ on $\partial \Omega$, such that $u_{n} \to v$ a.e. in $\partial \Omega$. \end{itemize} In this case, the set will be $\mathcal{T}^{1,p(\cdot)}_{tr}(\Omega)$ where tr is related to the trace of an element $u\in \mathcal{T}^{1,p(\cdot)}_{tr}(\Omega)$ (see \cite{a3,b2}). We can now introduce the notion of an entropy solution of \eqref{e1.1}. \begin{definition} \label{def4.2} \rm A measurable function $u$ is an entropy solution to problem \eqref{e1.1} if $ u \in \mathcal{T}^{1,p(\cdot)}_{\mathcal{H}}(\Omega)$, $b(u) \in L^{1}(\Omega)$ and for every $k > 0$, \begin{equation} \label{e4.1} \int_{\Omega}a(x,\nabla u).\nabla T_{k}(u-\varphi) dx + \int_{\Omega} b(u)T_{k}(u-\varphi) dx \leq \int_{\Omega}f(x)T_{k}(u-\varphi) dx, \end{equation} for all $\varphi \in W^{1,p(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$. \end{definition} Our main result in this section is the following. \begin{theorem} \label{thm4.3} Assume \eqref{e1.8}-\eqref{e1.12} and $f \in L^{1}(\Omega)$. Then there exists a unique entropy solution $u$ to \eqref{e1.1}. \end{theorem} To prove the above theorem, we need the following propositions among which, some can be proved following \cite{b3} with necessary changes in detail. But those which are new will be proved. \begin{proposition} \label{prop4.4} Assume \eqref{e1.8}-\eqref{e1.12}, $f \in L^{1}(\Omega)$ and $q(\cdot): \Omega \to [1, +\infty)$ a measurable function. Let $u$ be an entropy solution of \eqref{e1.1}. If there exists a positive constant $M$ such that \begin{equation} \label{e4.2} \int_{\{|u|>k\}} k^{q(x)} dx \leq M \quad\text{for all } k>0 \end{equation} then \[ \int_{\{| \nabla u|^{\alpha(\cdot)}>k\}} k^{q(x)} dx \leq C\|f\|_1 + M\quad \text{for all } k>0, \] where $\alpha(\cdot) = p(\cdot)/(q(\cdot)+1)$ and $C$ is a positive constant. \end{proposition} \begin{proposition} \label{prop4.5} Assume that \eqref{e1.8}-\eqref{e1.12} hold and $f\in L^{1}(\Omega)$. Let $u$ be an entropy solution of \eqref{e1.1}. Then \begin{equation} \label{e4.3} \int_{\Omega}|\nabla T_{k}(u)|^{p(x)}dx\leq C'k\|f\|_1 \quad \text{for all }k>0 \end{equation} and \begin{equation} \label{e4.4} \|b(u)\|_1\leq C''\operatorname{meas}(\Omega)\|f\|_1, \end{equation} where $C'$ and $C''$ are positive constants. \end{proposition} \begin{proposition} \label{prop4.6} Assume that \eqref{e1.8}-\eqref{e1.12} hold and $ f \in L^{1}(\Omega)$. Let $u$ be an entropy solution of \eqref{e1.1}. Then \begin{equation} \label{e4.5} \int_{\{ |u| \leq k \}} |\nabla T_{k}(u)|^{p_{-}} dx \leq C'''(k+1) \quad\text{for all } k > 0, \end{equation} where $C'''$ is a positive constant. \end{proposition} \begin{proposition} \label{prop4.7} Assume that \eqref{e1.8}-\eqref{e1.12} hold true and $ f \in L^{1}(\Omega)$. Let $u$ be an entropy solution of \eqref{e1.1}. Then \begin{equation} \label{e4.6} \operatorname{meas}\{ |u| > h \} \leq \frac{\|f\|_1} {\min (b(h),|b(-h)|)} \quad \text{for all $h$ large enough} \end{equation} and \begin{equation} \label{e4.7} \operatorname{meas}\{ | \nabla u| > h \} \leq \frac{\text{const}(\|f\|_1, p_{-} )}{h^{p_{-}-1}} \quad \text{for all } h \geq 1. \end{equation} \end{proposition} \begin{proof} We first prove \eqref{e4.6}. Indeed, by \eqref{e4.4} (see \cite[proof of (4.4)]{b3}, we have \[ \int_{\{|u|>h\}}|b(u)|dx\leq \|f\|_1. \] From this inequality, we deduce that \[ \min(b(h),|b(-h)|)\int_{\{|u|>h\}}dx\leq \|f\|_1. \] The proof of \eqref{e4.7} is similar to that of \cite[Proposition 4.8]{b3}. \end{proof} We remark that since $b$ is continuous and surjective, by \eqref{e4.6}, we deduce that \[ \operatorname{meas}\{ |u| > h \}\to 0 \quad\text{as }h\to+\infty. \] \subsection{Proof of Theorem \ref{thm4.3}} \textbf{Uniqueness of entropy solution.} Let $h>0$ and $u_1, u_2$ be two entropy solutions of \eqref{e1.1}. We write the entropy inequality \eqref{e4.1} corresponding to the solution $u_1$ with $T_{h}(u_2)$ as a test function and to the solution $u_2$ with $T_{h}(u_1)$ as a test function. Upon addition, we obtain \begin{equation} \label{e4.8} \begin{split} & \int_{\{|u_1-T_{h}(u_2)| \leq k \}} a(x,\nabla u_1). \nabla (u_1-T_{h}(u_2)) dx \\ & + \int_{\{|u_2-T_{h}(u_1)| \leq k \}} a(x,\nabla u_2). \nabla (u_2-T_{h}(u_1)) dx \\ & + \int_{\Omega} b(u_1) T_{k}(u_1-T_{h}(u_2)) dx + \int_{\Omega} b(u_2) T_{k}(u_2-T_{h}(u_1)) dx \\ &\leq \int_{\Omega} f(x)\Big( T_{k}(u_1-T_{h}(u_2)) + T_{k}(u_2-T_{h}(u_1)) \Big) dx . \end{split} \end{equation} Now define \[ E_1 := \{|u_1-u_2| \leq k, |u_2| \leq h \},\quad E_2 := E_1 \cap \{ |u_1| \leq h \}, \quad E_3 := E_1 \cap \{ |u_1| > h \}. \] We start with the first integral in \eqref{e4.8}. By \eqref{e1.12}, we have \begin{align} &\int_{\{|u_1-T_{h}(u_2)| \leq k \}} a(x,\nabla u_1). \nabla (u_1-T_{h}(u_2)) dx \nonumber \\ &= \int_{\{|u_1-T_{h}(u_2)| \leq k \} \cap \{|u_2| \leq h \}} a(x,\nabla u_1). \nabla (u_1-T_{h}(u_2)) dx \nonumber\\ &\quad + \int_{\{|u_1-T_{h}(u_2)| \leq k \} \cap \{|u_2| > h \}} a(x,\nabla u_1). \nabla (u_1-T_{h}(u_2)) dx \nonumber\\ &= \int_{\{|u_1-T_{h}(u_2)| \leq k \} \cap \{|u_2| \leq h \}} a(x,\nabla u_1). \nabla (u_1-u_2) dx \nonumber\\ &\quad+ \int_{\{|u_1- h\text{sign}(u_2)| \leq k \} \cap \{|u_2| > h \}} a(x,\nabla u_1). \nabla u_1 dx \nonumber\\ &\geq \int_{\{|u_1-T_{h}(u_2)| \leq k \} \cap \{|u_2| \leq h \}} a(x,\nabla u_1). \nabla (u_1-u_2) dx \label{e4.9} \\ & = \int_{E_1} a(x,\nabla u_1). \nabla (u_1-u_2) dx \nonumber\\ &= \int_{E_2} a(x,\nabla u_1). \nabla (u_1-u_2) dx + \int_{E_3} a(x,\nabla u_1). \nabla (u_1-u_2) dx \nonumber \\ &= \int_{E_2} a(x,\nabla u_1). \nabla (u_1-u_2) dx + \int_{E_3} a(x,\nabla u_1). \nabla u_1 dx - \int_{E_3} a(x,\nabla u_1). \nabla u_2 dx \nonumber\\ &\geq \int_{E_2} a(x,\nabla u_1). \nabla (u_1-u_2) dx - \int_{E_3} a(x,\nabla u_1). \nabla u_2 dx. \nonumber \end{align} Using \eqref{e1.10} and \eqref{e2.1}, we estimate the last integral in \eqref{e4.9} as follows. \begin{equation} \label{e4.10} \begin{split} &| \int_{E_3} a(x,\nabla u_1). \nabla u_2 dx |\\ & \leq C_1 \int_{E_3} (j(x) + |\nabla u_1|^{p(x)-1} ) |\nabla u_2| dx \\ &\leq C_1\Big( |j|_{p'(\cdot)} + | |\nabla u_1|^{p(x)-1} |_{p'(\cdot), \{ h < |u_1| \leq h+k\}}\Big) |\nabla u_2|_{p(\cdot), \{ h-k < |u_2| \leq h\}}, \end{split} \end{equation} where $\big| |\nabla u_1|^{p(x)-1} \big|_{p'(\cdot), \{ h < |u_1| \leq h+k\}} = \big\| |\nabla u_1|^{p(x)-1} \big\|_{L^{p'(\cdot)}(\{ h < |u_1| \leq h+k\})}$. Since $u_1$ is an entropy solution of \eqref{e1.1}, by taking $\varphi = T_{h}(u_1)$ in the entropy inequality \eqref{e4.1}, and using \eqref{e1.12}, we obtain \[ \int_{\{ h < |u_1| \leq h+k \}} |\nabla u_1|^{p(x)} dx \leq Ck \|f\|_1. \] So by Lemma \ref{lem2.1}, $$ \big| |\nabla u_1|^{p(x)-1} \big|_{p'(\cdot), \{ h < |u_1| \leq h+k\}} \leq C' < +\infty, $$ where $C'$ is a constant which does not depend on $h$. Therefore, $$ C_1( |j|_{p'(\cdot)} + | |\nabla u_1|^{p(x)-1} |_{p'(\cdot), \{ h < |u_1| \leq h+k\}}) \leq C_1\Big(|j|_{p'(\cdot)} + C' \Big) < +\infty. $$ Since $u_2$ is an entropy solution to problem \eqref{e1.1}, by taking $\varphi = T_{h}(u_2)$ in the entropy inequality \eqref{e4.1} and using \eqref{e1.12}, we obtain \[ \int_{\{ h < |u_2| \leq h+k \}} |\nabla u_2|^{p(x)} dx \leq Ck\int_{\{|u_2| > h \}} |f| dx. \] Using inequality \eqref{e4.6}, we have $\operatorname{meas}\{ |u_2| > h \} \to 0$ as $h \to + \infty$. As $f \in L^{1}(\Omega)$ we obtain \[ Ck\int_{\{|u_2| > h\}} |f| dx \to 0 \quad\text{as } h \to + \infty \text{ for any fixed number } k>0. \] From the above convergence we deduce that \[ \lim_{h \to + \infty} \int_{\{ h < |u_2| \leq h+k \}} |\nabla u_2|^{p(x)} dx = 0, \quad \text{for any fixed number } k>0. \] Hence \[ \lim_{h \to + \infty} \int_{\{ h-k < |u_2| \leq h \}} |\nabla u_2|^{p(x)} dx = \lim_{l \to + \infty} \int_{\{l < |u_2| \leq l+k \}} |\nabla u_2|^{p(x)} dx = 0, \] for any fixed $k>0$ with $l = h-k$. So by Lemma \ref{lem2.1}, $|\nabla u_2|_{p(\cdot), \{ h-k < |u_2| \leq h\}} \to 0$ as $h \to + \infty$, for any fixed number $k>0$. Therefore, from \eqref{e4.9} and \eqref{e4.10}, we obtain \begin{equation} \label{e4.11} \int_{\{|u_1-T_{h}(u_2)| \leq k \}} a(x,\nabla u_1). \nabla (u_1-T_{h}(u_2)) dx \geq I_{h} + \int_{E_2} a(x,\nabla u_1). \nabla (u_1-u_2) dx, \end{equation} where $I_{h}$ converges to zero as $h \to +\infty$. We may adopt the same procedure for study the second term in \eqref{e4.8} to obtain \begin{equation} \label{e4.12} \int_{\{|u_2-T_{h}(u_1)| \leq k \}} a(x,\nabla u_2). \nabla (u_2-T_{h}(u_1)) dx \geq J_{h} - \int_{E_2} a(x,\nabla u_2). \nabla (u_1-u_2) dx, \end{equation} where $J_{h}$ converges to zero as $h \to +\infty$. Now for all $h,k >0$, set \[ K_{h} = \int_{\Omega} b(u_1) T_{k}(u_1-T_{h}(u_2)) dx + \int_{\Omega} b(u_2) T_{k}(u_2-T_{h}(u_1)) dx. \] We have \[ b(u_1) T_{k}(u_1-T_{h}(u_2)) \to b(u_1) T_{k}(u_1-u_2) \quad\text{a.e. in $\Omega$ as } h \to +\infty \] and \[ | b(u_1) T_{k}(u_1-T_{h}(u_2)) | \leq k |b(u_1)| \in L^{1}(\Omega). \] Then by Lebesgue Theorem, we deduce that \begin{equation} \label{e4.13} \lim_{ h \to +\infty}\int_{\Omega} b(u_1) T_{k}(u_1-T_{h}(u_2)) dx = \int_{\Omega} b(u_1) T_{k}(u_1-u_2) dx. \end{equation} Similarly, we have \begin{equation} \label{e4.14} \lim_{ h \to +\infty}\int_{\Omega} b(u_2) T_{k}(u_2-T_{h}(u_1)) dx = \int_{\Omega} b(u_2) T_{k}(u_2-u_1) dx. \end{equation} Using \eqref{e4.13} and \eqref{e4.14}, we obtain \begin{equation} \label{e4.15} \lim_{ h \to +\infty}K_{h} = \int_{\Omega} (b(u_1)- b(u_2) ) T_{k}(u_1-u_2) dx. \end{equation} We next examine the right-hand side of \eqref{e4.8}. For all $k > 0$, \[ f(x)\Big( T_{k}(u_1-T_{h}(u_2)) + T_{k}(u_2-T_{h}(u_1)) \Big) \to f(x)\Big( T_{k}(u_1-u_2) + T_{k}(u_2-u_1) \Big) = 0 \] a.e. in $\Omega$ as $h \to +\infty$ and \[ | f(x)\Big( T_{k}(u_1-T_{h}(u_2)) + T_{k}(u_2-T_{h}(u_1)) \Big) | \leq 2k|f(x)| \in L^{1}(\Omega). \] Lebesgue Theorem allows us to write \begin{equation} \label{e4.16} \lim_{ h \to +\infty} \int_{\Omega} f(x)\Big( T_{k}(u_1-T_{h}(u_2)) + T_{k}(u_2-T_{h}(u_1)) \Big) dx =0. \end{equation} Using \eqref{e4.11}, \eqref{e4.12}, \eqref{e4.15} and \eqref{e4.16}, we obtain \begin{equation} \label{e4.17} \begin{split} &\int_{ \{|u_1- u_2| \leq k \}} \Big(a(x,\nabla u_1) - a(x,\nabla u_2) \Big). \Big(\nabla u_1- \nabla u_2 \Big) dx \\ &+ \int_{\Omega} (b(u_1)- b(u_2) ) T_{k}(u_1-u_2) dx \leq 0. \end{split} \end{equation} Therefore, \begin{equation} \label{e4.18} \int_{\Omega} (b(u_1)- b(u_2) ) T_{k}(u_1-u_2) dx=0, \end{equation} from which we deduce that \begin{equation} \label{e4.19} \lim_{k\to 0} \int_{\Omega}(b(u_1)- b(u_2))\frac{1}{k}T_k(u_1-u_2) \,dx = \int_{\Omega}|b(u_1)- b(u_2)| \,dx=0. \end{equation} It also follows from \eqref{e4.17} that \begin{equation} \label{e4.20} \int_{ \{|u_1- u_2| \leq k \}} \Big(a(x,\nabla u_1) - a(x,\nabla u_2) \Big). \Big(\nabla u_1- \nabla u_2 \Big) dx =0. \end{equation} Hence, from \eqref{e4.19} and \eqref{e4.20}, we obtain \begin{gather*} u_1 - u_2=c \quad \text{a.e. in } \Omega.\\ \text{and } b(u_1)=b(u_2). \end{gather*} \textbf{Existence of entropy solution.} Let $f_{n}=T_{n}(f)$; then $\{f_{n}\}_{n=1}^{+\infty}$ is a sequence of bounded functions which strongly converges to $f\in L^{1}(\Omega)$ and is such that \begin{equation} \label{e4.21} \|f_{n}\|_1\leq\|f\|_1 , \quad \text{for all }n\in \mathbb{N}. \end{equation} We consider the problem \begin{equation} \label{e4.22} \begin{gathered} -\operatorname{div} a(x,\nabla u_{n}) + b(u_{n}) = f_{n} \quad \text{in } \Omega,\\ a(x,\nabla u_n).\eta = 0 \quad \text{on }\partial \Omega. \end{gathered} \end{equation} It follows from Theorem \ref{thm3.2} that there exists a unique function $u_{n} \in W^{1,p(\cdot)}(\Omega)$ such that \begin{equation} \label{e4.23} \int_{\Omega} a(x,\nabla u_{n}).\nabla \varphi dx + \int_{\Omega} b(u_{n}) \varphi dx = \int_{\Omega} f_{n} \varphi dx \end{equation} for all $\varphi \in W^{1,p(\cdot)}(\Omega)$. Our aim is to prove that these approximated solutions $u_{n}$ tend, as $n$ goes to infinity, to a measurable function $u$ which is an entropy solution to the limit problem \eqref{e1.1}. To start with, we prove the following lemma. \begin{lemma} \label{lem4.8} For any $k > 0$, $$ \| T_{k}(u_{n})\|_{1,p(\cdot)} \leq 1 + C, $$ where $C = C(C_3,k, f, p_{-}, p_{+}, \operatorname{meas}(\Omega))$ is a positive constant. \end{lemma} \begin{proof} By taking $\varphi = T_{k}(u_{n})$ in \eqref{e4.23}, we obtain \[ \int_{\Omega} a(x,\nabla u_{n}).\nabla T_{k}(u_{n})dx +\int_{\Omega} b(u_{n}) T_{k}(u_{n})dx = \int_{\Omega} f_{n} T_{k}(u_{n}) dx. \] Since all the terms in the left-hand side of equality above are nonnegative and \[ \int_{\Omega} f_{n} T_{k}(u_{n}) dx \leq k \|f_{n}\|_1 \leq k \|f\|_1, \] by using \eqref{e1.12} we obtain \begin{equation} \label{e4.24} \int_{\Omega} |\nabla T_{k}(u_{n})|^{p(x)} dx \leq Ck \|f\|_1. \end{equation} We also have that \[ \int_{\Omega} |T_{k}(u_{n})|^{p(x)} dx = \int_{ \{|u_{n}| \leq k \}} |T_{k}(u_{n})|^{p(x)} dx + \int_{ \{ |u_{n}| > k \}} |T_{k}(u_{n})|^{p(x)}dx. \] Furthermore, \[ \int_{\{|u_{n}| > k \}} |T_{k}(u_{n})|^{p(x)} dx = \int_{\{|u_{n}| > k \}} k^{p(x)} dx \leq \begin{cases} k^{p_{+}}\operatorname{meas}(\Omega) &\text{if } k \geq 1,\\ \operatorname{meas}(\Omega) &\text{if } k < 1 \end{cases} \] and \[ \int_{\{|u_{n}|\leq k \}} |T_{k}(u_{n})|^{p(x)} dx \leq \int_{\{|u_{n}| \leq k \}} k^{p(x)} dx \leq \begin{cases} k^{p_{+}}\operatorname{meas}(\Omega) &\text{if } k \geq 1,\\ \operatorname{meas}(\Omega) &\text{if } k < 1. \end{cases} \] This allows us to write \begin{equation} \label{e4.25} \int_{\Omega} |T_{k}(u_{n})|^{p(x)} dx \leq 2(1+k^{p_{+}})\operatorname{meas}(\Omega). \end{equation} Hence, adding \eqref{e4.24} and \eqref{e4.25} yields \begin{equation} \label{e4.26} \rho_{1,p(\cdot)} (T_{k}(u_{n})) \leq Ck\|f\|_1 + (1+k^{p_{+}})\operatorname{meas}(\Omega) = C(C_3,k, f, p_{+}, \operatorname{meas}(\Omega)). \end{equation} For $\|T_{k}(u_{n})\|_{1,p(\cdot)} \geq 1$, we have \[ \|T_{k}(u_{n})\|^{p_{-}}_{1,p(\cdot)} \leq \rho_{1,p(\cdot)} (T_{k}(u_{n})) \leq C(C_3,k, f, p_{+}, \operatorname{meas}(\Omega)), \] which is equivalent to \[ \|T_{k}(u_{n})\|_{1,p(\cdot)} \leq \Big(C(C_3,k, f, p_{+}, \operatorname{meas}(\Omega)) \Big)^{1/p_{-}} = C(C_3,k, f, p_{-}, p_{+}, \operatorname{meas}(\Omega)). \] The above inequality gives \[ \|T_{k}(u_{n})\|_{1,p(\cdot)} \leq 1+ C(C_3,k, f, p_{-}, p_{+}, \operatorname{meas}(\Omega)). \] The proof is complete. \end{proof} From Lemma \ref{lem4.8} we deduce that for any $k >0$, the sequence $\{T_{k}(u_{n})\}_{n=1}^{+\infty}$ is uniformly bounded in $ W^{1,p(\cdot)}(\Omega)$ and so in $W^{1,p_{-}}(\Omega)$. Then, up to a subsequence we can assume that for any $k>0$, $T_{k}(u_{n})$ converges weakly to $\sigma_{k}$ in $W^{1,p_{-}}(\Omega)$, and so $T_{k}(u_{n})$ strongly converges to $\sigma_{k}$ in $L^{p_{-}}(\Omega)$. \begin{proposition} \label{prop4.9}. Assume that \eqref{e1.8}-\eqref{e1.12} hold and $u_{n} \in W^{1,p(\cdot)}(\Omega) $ is the solution of \eqref{e4.22}. Then the sequence $\{u_{n}\}_{n=1}^{+\infty}$ is Cauchy in measure. In particular, there exists a measurable function $u$ and a subsequence still denoted $\{u_{n}\}_{n=1}^{+\infty}$ such that $ u_{n} \to u $ in measure. \end{proposition} \begin{proof} Let $s >0$ and $k > 0$ be fixed. Define \[ E_{n}:= \{ |u_{n}| > k\},\quad E_{m}:= \{ |u_{m}| > k\},\quad E_{n,m}:= \{ |T_{k}(u_{n}) - T_{k}(u_{m})| > s\}\,. \] Note that \[ \{ |u_{n} - u_{m}| > s\} \subset E_{n} \cup E_{m} \cup E_{n,m} \] and hence \begin{equation} \label{e4.27} \operatorname{meas} \{ |u_{n} - u_{m}| > s\} \leq \operatorname{meas} (E_{n}) + \operatorname{meas} (E_{m}) + \operatorname{meas} (E_{n,m}). \end{equation} Let $\epsilon >0$. Using Proposition \ref{prop4.7}, we choose $k = k(\epsilon)$ such that \begin{equation} \label{e4.28} \operatorname{meas} (E_{n}) \leq \epsilon/3 \quad \text{and}\quad \operatorname{meas} (E_{m}) \leq \epsilon/3. \end{equation} Since $T_{k}(u_{n})$ converges strongly in $L^{p_{-}}(\Omega)$, then it is a Cauchy sequence in $L^{p_{-}}(\Omega)$. Thus \begin{equation} \label{e4.29} \operatorname{meas}(E_{n,m}) \leq \frac{1}{s^{p_{-}}} \int_{\Omega} |T_{k}(u_{n}) - T_{k}(u_{m})|^{p_{-}} dx \leq \frac{\epsilon}{3}, \end{equation} for all $n,m \geq n_0(s,\epsilon)$. Finally, from \eqref{e4.27}, \eqref{e4.28} and \eqref{e4.29}, we obtain \begin{equation} \label{e4.30} \operatorname{meas} \{ |u_{n} - u_{m}| > s\} \leq \epsilon \quad \text{ for all } n,m \geq n_0(s,\epsilon). \end{equation} Relations \eqref{e4.30} imply that the sequence $\{u_{n}\}_{n=1}^{+\infty}$ is a Cauchy sequence in measure and the proof is complete. \end{proof} Note that as $ u_{n} \to u $ in measure, up to a subsequence, we can assume that $ u_{n} \to u $ a. e. in $\Omega$. In the sequel, we need the following two technical lemmas (see \cite{h1,s1}). \begin{lemma} \label{lem4.10} Let $\{v_{n}\}_{n=1}^{+\infty}$ be a sequence of measurable functions in $\Omega$. If $v_{n}$ converges in measure to $v$ and is uniformly bounded in $L^{p(\cdot)}(\Omega)$ for some $1 \ll p(\cdot) \in L^{\infty}(\Omega)$, then $v_{n} \to v$ strongly in $L^{1}(\Omega)$. \end{lemma} The second technical lemma is a well known result in the measure theory \cite{h1}. \begin{lemma} \label{lem4.11} Let $(X, \mathcal{M}, \mu)$ be a measure space such that $\mu(X) < +\infty$. Consider a measurable function $\gamma : X \to [0, +\infty]$ such that \[ \mu( \{ x \in X : \gamma(x) = 0 \}) = 0. \] Then, for every $\epsilon > 0$, there exists $\delta > 0$, such that \[ \mu(A) < \epsilon, \quad \text{for all $A \in \mathcal{M}$ with } \int_{A} \gamma d \mu < \delta. \] \end{lemma} We are ready for proving that the function $u$ in the Proposition \ref{prop4.9} is an entropy solution of \eqref{e1.1}. Let $\varphi \in W^{1,p(\cdot)}(\Omega) \cap L^{\infty}(\Omega)$. For any $k>0$, choose $T_{k}(u_{n}- \varphi)$ as a test function in \eqref{e4.23}. We obtain \begin{equation} \label{e4.31} \begin{split} &\int_{\Omega} a(x,\nabla u_{n}). \nabla T_{k}(u_{n} - \varphi) dx + \int_{\Omega} b(u_{n}) T_{k}(u_{n} - \varphi) dx\\ &= \int_{\Omega} f_{n}(x) T_{k}(u_{n} - \varphi)dx. \end{split} \end{equation} The following proposition is useful to pass to the limit in the first term of \eqref{e4.31}. \begin{proposition} \label{prop4.12} Assume that \eqref{e1.8}--\eqref{e1.12} hold and $u_{n} \in W^{1,p(\cdot)}(\Omega) $ is the weak solution to \eqref{e4.22}. Then \begin{itemize} \item[(i)] $\nabla u_{n}$ converges in measure to the weak gradient of $u$; \item[(ii)] For all $k>0$, $\nabla T_{k}(u_{n})$ converges to $\nabla T_{k}(u)$ in $(L^{1}(\Omega))^{N}$. \item[(iii)] For all $t>0$, $a(x,\nabla T_{t}(u_{n}))$ converges strongly to $a(x,\nabla T_{t}(u))$ in $(L^{1}(\Omega))^{N}$ and weakly in $(L^{p'(\cdot)}(\Omega))^{N}$. \end{itemize} \end{proposition} \begin{proof} (i) We claim that the sequence $\{\nabla u_{n}\}_{n=1}^{+\infty}$ is Cauchy in measure. Indeed, let $s>0$ and consider \[ A_{n,m}:=\{ |\nabla u_{n}| >h \} \cup \{ |\nabla u_{m}| >h \},\quad B_{n,m}:=\{ | u_{n} - u_{m}| >k \} \] and \[ C_{n,m}:=\{ |\nabla u_{n}| \leq h, |\nabla u_{m}| \leq h,\, | u_{n} - u_{m}| \leq k,\;|\nabla u_{n} - \nabla u_{m}| > s\}, \] where $h$ and $k $ will be chosen later. Note that \begin{equation} \label{e4.32} |\nabla u_{n} - \nabla u_{m}| > s \} \subset A_{n,m} \cup B_{n,m} \cup C_{n,m}. \end{equation} Let $\epsilon >0$. By Proposition \ref{prop4.7} (relation \eqref{e4.7}), we may choose $h = h(\epsilon)$ large enough such that \begin{equation} \label{e4.33} \operatorname{meas}(A_{n,m}) \leq \epsilon / 3, \end{equation} for all $n, m \geq 0$. On the other hand, by Proposition \ref{prop4.9}, \begin{equation} \label{e4.34} \operatorname{meas}(B_{n,m}) \leq \epsilon / 3, \end{equation} for all $n, m \geq n_0(k,\epsilon)$. Moreover, since $a(x,\xi)$ is continuous with respect to $\xi$ for a.e. $x \in \Omega$, by assumption \eqref{e1.11} there exists a real valued function $\gamma : \Omega \to [0, +\infty]$ such that $\operatorname{meas}(\{ x \in \Omega: \gamma (x) = 0\}) = 0$ and \begin{equation} \label{e4.35} (a(x,\xi) - a(x,\xi')).(\xi - \xi') \geq \gamma(x), \end{equation} for all $\xi, \xi' \in \mathbb{R}^{N}$ such that $|\xi| \leq h,\, |\xi'| \leq h,\,|\xi - \xi'| \geq s$, for a.e. $x \in \Omega$. Let $ \delta = \delta(\epsilon)$ be given by Lemma \ref{lem4.11}, replacing $ \epsilon$ and $A$ by $\epsilon/3$ and $C_{n,m}$ respectively. As $u_{n} $ is a weak solution of \eqref{e4.22}, using $T_{k}(u_{n} - u_{m})$ as a test function in \eqref{e4.23}, we obtain \begin{align*} &\int_{\Omega} a(x,\nabla u_{n}).\nabla T_{k}(u_{n} - u_{m}) dx + \int_{\Omega} b(u_{n}) T_{k}(u_{n} - u_{m})dx\\ &= \int_{\Omega} f_{n} T_{k}(u_{n} - u_{m}) dx \leq k\|f\|_1. \end{align*} Similarly for $u_{m}$, we have \begin{align*} &\int_{\Omega} a(x,\nabla u_{m}).\nabla T_{k}(u_{m} - u_{n}) dx + \int_{\Omega} b(u_{m}) T_{k}(u_{m} - u_{n})dx\\ &= \int_{\Omega} f_{m} T_{k}(u_{m} - u_{n}) dx \leq k\|f\|_1. \end{align*} Adding these two inequalities yields \begin{align*} & \int_{\{ |u_{n} -u_{m}| \leq k \}} ( a(x,\nabla u_{n}) - a(x,\nabla u_{m})).(\nabla u_{n} - \nabla u_{m}) dx \\ & + \int_{\Omega} \Big(b(u_{n}) - b(u_{m}) \Big)T_{k}(u_{n} - u_{m})dx \leq 2k\|f\|_1. \end{align*} Since the second term of the above inequality is nonnegative, by using \eqref{e4.35} we obtain \[ \int_{C_{n,m}} \gamma(x) dx \leq \int_{C_{n,m}} ( a(x,\nabla u_{n}) - a(x,\nabla u_{m})).(\nabla u_{n} - \nabla u_{m}) dx \leq 2k\|f\|_1 < \delta, \] where $ k = \delta / 4\|f\|_1$. From Lemma \ref{lem4.11}, it follows that \begin{equation} \label{e4.36} \operatorname{meas}(C_{n,m}) \leq \epsilon / 3. \end{equation} Thus, using \eqref{e4.32}, \eqref{e4.33}, \eqref{e4.34} and \eqref{e4.36}, we obtain \begin{equation} \label{e4.37} \operatorname{meas}( \{|\nabla u_{n} - \nabla u_{m}| > s \}) \leq \epsilon, \quad \text{for all } n,m \geq n_0(s, \epsilon) \end{equation} and then the claim is proved. Consequently, $\{\nabla u_{n}\}_{n=1}^{+\infty}$ converges in measure to some measurable function $v$. To complete the proof of (i), we need the following lemma. \begin{lemma} \label{lem4.13} \begin{itemize} \item[(a)] For a.e. $t\in\mathbb{R}$, $\nabla T_{t}(u_{n})$ converges in measure to $v\chi_{\{|u|0$, \begin{align*} &\{|\chi_{\{|u_{n}|\delta\}\\ &\subset \{|\chi_{\{|u_{n}|\delta\}\\ &\leq \operatorname{meas}\{|u|=t\}+ \operatorname{meas}\{u_{n} h\} \] for all $h>0$. Due to Proposition \ref{prop4.9}, for all fixed $h>0$, we have $\operatorname{meas}\{|u-u_{n}|> h\}\to 0$ as $n\to +\infty$. Since $\operatorname{meas}\{t0$, one can find $N$ such that for all $n>N$, $\operatorname{meas}\{u_{n}\delta\}\to 0$ as $n\to +\infty$. Finally, since $\nabla T_{t}(u_{n})=\nabla u_{n}\chi_{\{|u_{n}|0$, $k>0$ and consider \begin{gather*} F_{n,m}=\{|\nabla u_{n}-\nabla u_{m}|>s,|u_{n}| \leq k,|u_{m}|\leq k\}, \\ G_{n,m}=\{|\nabla u_{m}|>s,|u_{n}|> k,|u_{m}|\leq k\},\\ H_{n,m}=\{|\nabla u_{n}|>s,|u_{m}|> k,|u_{n}|\leq k\},\quad I_{n,m}=\{0>s,|u_{m}|> k,|u_{n}|> k\}. \end{gather*} Note that \begin{equation} \label{e4.39} \{|\nabla T_{k}(u_{n})-\nabla T_{k}(u_{m})|>s\} \subset F_{n,m}\cup G_{n,m} \cup H_{n,m}\cup I_{n,m}. \end{equation} Let $\epsilon>0$. By Proposition \ref{prop4.7}, we may choose $k(\epsilon)$ such that \begin{equation} \label{e4.40} \operatorname{meas}(G_{n,m})\leq \frac{\epsilon}{4},\text{ meas}(H_{n,m})\leq \frac{\epsilon}{4} \text{ and }\operatorname{meas}(I_{n,m})\leq \frac{\epsilon}{4}. \end{equation} Therefore, using \eqref{e4.37}, \eqref{e4.39} and \eqref{e4.40}, we obtain \begin{equation} \label{e4.41} \operatorname{meas}( \{|\nabla T_{k}(u_{n}) - \nabla T_{k}(u_{m})| > s \}) \leq \epsilon, \quad \text{ for all } n,m \geq n_1(s, \epsilon). \end{equation} Consequently, $\nabla T_{k}(u_{n})$ converges in measure to $\nabla T_{k}(u)$. Then, using lemmas \ref{lem4.8} and \ref{lem4.10}, (ii) follows. \smallskip (iii) By lemmas \ref{lem4.10} and \ref{lem4.13}, for all $t>0$, $a(x,\nabla T_{t}(u_{n}))$ converges strongly to $a(x,\nabla T_{t}(u))$ in $(L^{1}(\Omega))^{N}$, and $a(x,\nabla T_{t}(u_{n}))$ converges weakly to $\chi_{t}\in (L^{p'(\cdot)}(\Omega))^{N}$ in $(L^{p'(\cdot)}(\Omega))^{N}$. Since each of the convergence implies the weak $L^{1}$-convergence, $\chi_{t}$ can be identified with $a(x,\nabla T_{t}(u))$; thus, $a(x,\nabla T_{t}(u))\in (L^{p'(\cdot)}(\Omega))^{N}$. The proof of (iii) is then complete. Thus the proof is complete. \end{proof} We are now able to pass to the limit in the identity \eqref{e4.31}. For the right-hand side, the convergence is obvious since $f_{n}$ converges strongly to $f$ in $L^{1}(\Omega)$ and $T_{k}(u_{n}- \varphi)$ converges weakly-$\ast$ to $ T_{k}(u - \varphi)$ in $L^{\infty}(\Omega)$ and a.e. in $\Omega$. For the second term of \eqref{e4.31}, we have $$ \int_{\Omega} b(u_{n}) T_{k}(u_{n} - \varphi) dx = \int_{\Omega}(b(u_{n}) - b(\varphi ))T_{k}(u_{n} - \varphi) dx + \int_{\Omega} b(\varphi) T_{k}(u_{n} - \varphi) dx. $$ The quantity $ (b(u_{n}) - b(\varphi) )T_{k}(u_{n} - \varphi) $ is nonnegative and since for all $s \in \mathbb{R}$, $s \mapsto b(s)$ is continuous, we obtain \[ (b(u_{n}) - b(\varphi) )T_{k}(u_{n} - \varphi) \to (b(u) - b(\varphi) )T_{k}(u - \varphi) \quad \text{ a.e. in } \Omega. \] Then, it follows by Fatou's Lemma that \begin{equation*} \liminf_{n \to +\infty} \int_{\Omega}(b(u_{n}) - b(\varphi) )T_{k}(u_{n} - \varphi) dx \geq \int_{\Omega}(b(u) - b(\varphi) )T_{k}(u - \varphi) dx. \end{equation*} We have $b(\varphi) \in L^{1}(\Omega)$. Since $T_{k}(u_{n}- \varphi)$ converges weakly-$\ast$ to $ T_{k}(u - \varphi)$ in $L^{\infty}(\Omega)$ and $ b(\varphi) \in L^{1}(\Omega)$, it follows that \[ \lim_{n \to +\infty} \int_{\Omega} b(\varphi) T_{k}(u_{n}- \varphi) dx = \int_{\Omega} b(\varphi) T_{k}(u - \varphi) dx. \] Next, we write the first term in \eqref{e4.31} in the form \begin{equation} \label{e4.42} \int_{ \{|u_{n}-\varphi| \leq k \}} a(x,\nabla u_{n}). \nabla u_{n} dx - \int_{ \{|u_{n}-\varphi| \leq k \}} a(x,\nabla u_{n}). \nabla \varphi dx. \end{equation} Set $l = k + \|\varphi\|_{\infty}$. The second integral in \eqref{e4.42} is equal to \[ \int_{ \{|u_{n}-\varphi| \leq k \}} a( x,\nabla T_{l}(u_{n})). \nabla \varphi dx. \] Since $a( x,\nabla T_{l}(u_{n}))$ is uniformly bounded in $(L^{p'(\cdot)}(\Omega))^{N}$ (by \eqref{e1.10} and \eqref{e4.24}), by Proposition \ref{prop4.12}-(iii), it converges weakly to $ a( x,\nabla T_{l}(u)) $ in $(L^{p'(\cdot)}(\Omega))^{N}$. Therefore, \[ \lim_{ n \to +\infty} \int_{ \{|u_{n}-\varphi| \leq k \}} a( x,\nabla T_{l}(u_{n})). \nabla \varphi dx = \int_{ \{|u-\varphi| \leq k \}} a( x,\nabla T_{l}(u)). \nabla \varphi dx. \] Moreover, $ a(x,\nabla u_{n}). \nabla u_{n} $ is nonnegative and converges a.e. in $\Omega$ to $ a(x,\nabla u). \nabla u $. Thanks to Fatou's Lemma, we obtain \begin{equation*} \liminf_{ n \to +\infty} \int_{ \{|u_{n}-\varphi| \leq k \}} a(x,\nabla u_{n}). \nabla u_{n} dx \geq \int_{ \{|u-\varphi| \leq k \}} a(x,\nabla u ). \nabla u dx. \end{equation*} Gathering results, we obtain \[ \int_{ \Omega} a(x,\nabla u ). \nabla T_{k}(u - \varphi ) dx + \int_{\Omega} b(u)T_{k}(u - \varphi) dx \leq \int_{\Omega} f T_{k}(u - \varphi) dx . \] We conclude that $u$ is an entropy solution of \eqref{e1.1}. \end{proof} \subsection*{Acknowledgments} The authors want to express their gratitude to the editor and the anonymous referees for comments and suggestions on the paper. \begin{thebibliography}{00} \bibitem{a1} A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti; \emph{Existence results for non-linear elliptic equations with degenerate coercivity}, Ann. Mat. Pura Appl. \textbf{182} (2003), 53-79. \bibitem{a2} F. Andreu, N. Igbida, J. M. Maz\'on, J. 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