\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 92, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/92\hfil
Entropy solutions for  $p(x)$-Laplace equations]
{Entropy solutions for nonlinear elliptic equations with variable exponents}

\author[C. Zhang \hfil EJDE-2014/92\hfilneg]
{Chao Zhang}  % in alphabetical order

\address{Chao Zhang \newline
Department of Mathematics\\
Harbin Institute of Technology\\
Harbin 150001, China}
\email{czhangmath@hit.edu.cn}

\thanks{Submitted August 26, 2013. Published March 4, 2014.}
\subjclass[2000]{35J70, 35D05, 35D10, 46E35}
\keywords{Variable exponents; entropy solutions; existence; uniqueness}

\begin{abstract}
 In this article we prove the existence and uniqueness of entropy solutions
 for $p(x)$-Laplace equations with a Radon measure which is absolutely
 continuous with respect to the relative $p(x)$-capacity.
 Moreover, the existence of entropy solutions for weighted $p(x)$-Laplace
 equation is also obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The study of partial differential equations and variational problems with 
non-standard growth conditions has been received considerable attention by
 many models coming from various branches of mathematical physics, such as 
elastic mechanics, image processing and electro-rheological fluid
dynamics, etc. We refer the readers to \cite{CLR,D,RR,R} and references therein.

Let $\Omega$ be a bounded open domain in $\mathbb{R}^N$ $(N\ge 2)$
with Lipschitz boundary $\partial\Omega$. In this article we consider the 
nonlinear elliptic problem
\begin{equation}\label{emain}
\begin{gathered}
-\operatorname{div} \big(w(x)|\nabla u|^{p(x)-2}\nabla u\big)=f \quad
\text{in } \ \Omega,\\
u=0 \quad \text{on }  \partial\Omega,
\end{gathered}
\end{equation}
where the variable exponent $p: \overline{\Omega}\to (1,\infty)$ is a
continuous function, $w$ is a weight function and $f\in L^1(\Omega)$.

When dealing with the $p$-Laplacian type equations with $L^1$ or measure data, 
it is reasonable to work with entropy solutions or renormalized solutions, 
which need less regularity than the usual weak solutions.
The notion of entropy solutions has been proposed by B\'{e}nilan et al. in \cite{B} 
for the nonlinear elliptic
problems. This framework was extended to related problems with constant $p$ 
in \cite{AB,BGO,C,P} and variable
exponents $p(x)$ in \cite{BW,RSU,SU,ZZ}. The interesting and difficult cases 
are those of $1<p\le N$, since the
variational methods of Leray-Lions (see \cite{LI}) can be easily applied for $p>N$.

Recently, when $w(x)\equiv 1$, the existence and uniqueness of entropy solutions 
of $p(x)$-Laplace equation
with $L^1$ data were proved in \cite{SU} by Sanch\'{o}n and Urbano.
 The proofs rely crucially on {\it a priori} estimates in
Marcinkiewicz spaces with variable exponents. Moreover, in \cite{ZZ} we 
extended the results in \cite{SU} to the case of a signed measure $\mu$ 
in $L^1(\Omega)+W^{-1, p'(\cdot)}(\Omega)$. In view of a refined method which 
is slightly different from \cite{SU}, we obtained that the entropy solution 
of problem \eqref{emain} is also a renormalized solution and proved the uniqueness 
of entropy solutions and renormalized solutions, and thus the
equivalence of entropy solutions and renormalized solutions. Especially, 
when $p$ is a constant function, $w$ is an $A_p$ weight and $f\in L^1(\Omega)$, 
Cavalheiro in \cite{C} proved the existence of entropy solutions for the 
Dirichlet problem \eqref{emain}.


This work is a natural extension of the results in \cite{C,ZZ}. 
The novelties in this paper are mainly two parts. First, when $p$ is a constant 
function, we know from \cite{BGO} that $\mu\in L^1(\Omega)+W^{-1,p'}(\Omega)$ 
if and only if $\mu\in \mathcal{M}^{p}_b(\Omega)$, i.e., every signed measure 
that is zero on the sets of zero $p$-capacity can be decomposed into the sum 
of a function in $L^1(\Omega)$ and an element in $W^{-1,p'}(\Omega)$, and
conversely, every signed measure in $L^1(\Omega)+W^{-1,p'}(\Omega)$ has 
zero measure for the sets of zero $p$-capacity. In our previous paper \cite{ZZ}, 
we proposed an open problem: what about the similar decomposition result 
for the variable exponent case? By using the similar arguments as in \cite{BGO} 
and employing the properties of $L^{p(\cdot)}(\Omega)$ and the relative 
$p(\cdot)$-capacity (see \cite{HHK}), we try to give a positive answer for 
this question. Although the proof follows basically the steps in \cite{BGO}, 
it is not a straightforward generalization of the same result for constant 
exponents which needs a more careful analysis to derive the conclusion. 
Second, as far as we know, there are no papers concerned with the entropy 
solutions for the weighted $p(x)$-Laplace equations. The main difficulty 
is that there are few results for the $A_{p(\cdot)}$-weight whenever $p$ 
is not constant function. We refer the readers to paper \cite{HD} 
by H\"{a}st\"{o} and Diening for the latest results. The properties of 
weighted variable exponent Lebesgue-Sobolev spaces in \cite{HD,KWZ} provide a 
way to prove the existence of entropy solutions for problem \eqref{emain}.

Now we review the definitions and basic properties of the weighted generalized 
Lebesgue spaces
$L^{p(x)}(\Omega,w)$ and weighted generalized Lebesgue-Sobolev spaces 
$W^{k,p(x)}(\Omega,w)$.

Let $w$ be a measurable positive and a.e. finite function in $\mathbb{R}^N$.
Set $C_+(\overline\Omega)=\{h\in C(\overline\Omega):
\min_{x\in\overline\Omega}h(x)>1\}$. For any $h\in C_+(\overline\Omega)$ we define
$$
h_+=\sup_{x\in \Omega}h(x)\quad \text{and}\quad
h_-=\inf_{x\in \Omega}h(x).
$$
For any $p\in C_+(\overline\Omega)$, we introduce the weighted variable
exponent Lebesgue space $L^{p(\cdot)}(\Omega,w)$ to consist of all
measurable functions such that
$$
\int_\Omega w(x)|u(x)|^{p(x)}\,dx<\infty,
$$
endowed with the Luxemburg norm
$$
\|u\|_{L^{p(x)}(\Omega,w)}=\inf\big\{\lambda >0: \int_\Omega
w(x)\Big|\frac{u(x)}{\lambda}\Big|^{p(x)}\,dx\le1\big\}.
$$

For any positive integer $k$, denote
$$
W^{k,p(x)}(\Omega,w)=\{u\in L^{p(x)}(\Omega,w): D^\alpha u\in
L^{p(x)}(\Omega,w), |\alpha|\le k\},
$$
with the norm
$$
\|u\|_{W^{k,p(x)}(\Omega,w)}
=\sum_{|\alpha|\le k}\|D^{\alpha}u\|_{L^{p(x)}(\Omega,w)}.
$$
An interesting feature of a generalized Lebesgue-Sobolev 
space is that smooth functions are not dense in it without additional 
assumptions on the exponent $p(x)$. This was observed by Zhikov \cite{Z} 
in connection with Lavrentiev phenomenon. However, when the exponent $p(x)$
 is {\it log-H\"{o}lder} continuous, i.e., there is a constant $C$ such that
\begin{align}\label{assume}
|p(x)-p(y)|\le \frac{C}{-\log|x-y|}
\end{align}
for every $x, y\in \Omega$ with $|x-y|\le 1/2$, then smooth functions are 
dense in variable exponent Sobolev spaces and there is no confusion in defining 
the Sobolev space with zero boundary values, $W^{1,p(\cdot)}_0(\Omega)$, as the 
completion of $C_0^\infty(\Omega)$ with respect to the norm 
$\|u\|_{W^{1,p(\cdot)}(\Omega)}$ (see \cite{H}).

Let $T_k$ denote the truncation function at height $k\ge 0$:
\[
T_k(r)=\min\{k, \max\{r,-k\}\}
=\begin{cases} 
k & \text{if } r\ge k,\\
r & \text{if } |r|<k,\\
-k & \text{if }r\le -k.
\end{cases} 
\]
Denote
\[
\mathcal{T}_0^{1,p(\cdot)}(\Omega)
=\{u:u \text{ is measurable, } T_k(u)\in W_0^{1,p(\cdot)}(\Omega,w), 
\text{ for every }k>0\}.
\]

Next we define the very weak gradient of a measurable function
$u\in\mathcal {T}_0^{1,p(\cdot)}(\Omega)$. As a matter of the fact, 
working as in \cite[Lemma 2.1]{B}, we
have the following result.

\begin{proposition}\label{prop1}
For every function  $u\in\mathcal {T}_0^{1,p(\cdot)}(\Omega)$, 
there exists a unique measurable function $v: \Omega\to \mathbb{R}^N$,
which we call the very weak gradient of $u$ and denote $v=\nabla u$,  such that
$$
\nabla T_k(u)=v\chi_{\{|u|<k\}} \quad\text{ for a.e. $x\in \Omega$
 and for every } k>0,
$$
where $\chi_E$ denotes the characteristic function of a measurable
set $E$. Moreover, if u belongs to $W^{1,1}_0(\Omega,w)$, then $v$
coincides with the weak gradient of $u$.
\end{proposition}

The notion of the very weak gradient allows us to give the following
definition of entropy solutions for problem \eqref{emain}.

\begin{definition} \label{def1} \rm
A function $u\in \mathcal {T}_0^{1,p(\cdot)}(\Omega)$ is called
an entropy solution to problem \eqref{emain} if
\begin{equation} \label{edef-inequality}
\int_\Omega w(x)|\nabla u|^{p(x)-2}\nabla u\cdot \nabla
T_k(u-\phi)\,dx=\int_\Omega f T_k(u-\phi)\,dx,
\end{equation}
for all $\phi\in W_0^{1,p(x)}(\Omega,w)\cap L^\infty(\Omega)$.
\end{definition}


The rest of this paper is organized as follows. 
In Section $2$, we prove the existence and uniqueness of entropy solutions 
for $p(x)$-Laplace equation with a Radon measure which is absolutely 
continuous with respect to the relative $p(\cdot)$-capacity. 
The existence of entropy solutions for weighted $p(x)$-Laplace equation 
will be considered in Section $3$. In the following sections
$C$ will represent a generic constant that may change from line to
line even if in the same inequality.


\section{Unweighted case}

In this section, we prove the existence and uniqueness of entropy solutions 
for the following problem
\begin{equation}\label{eprob1}
\begin{gathered} 
-\operatorname{div} \big(|\nabla u|^{p(x)-2}\nabla u\big)=\mu \quad\text{in } \Omega,\\
u=0 \quad \text{on }  \partial\Omega,
\end{gathered}
\end{equation}
where $\mu$ a Radon measure which is absolutely continuous with respect 
to the relative $p(\cdot)$-capacity.
First we state some results that will be used later.

\begin{lemma}[\cite{FZ, KR}] \label{lem1}
{\rm (1)} The space $L^{p(\cdot)}(\Omega)$ is a
separable, uniform convex Banach space, and its conjugate space is
$L^{p'(\cdot)}(\Omega)$ where $1/p(x)+1/p'(x)=1$. For any $u\in
L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$, we have
$$
\big|\int_\Omega uv\,dx\big|\le
\big(\frac{1}{p_{-}}+\frac{1}{(p_{-})'}\big)\|u\|_{L^{p(x)}(\Omega)}\|v\|_{L^{p'(x)}(\Omega)}\le
2\|u\|_{L^{p(x)}(\Omega)}\|v\|_{L^{p'(x)}(\Omega)};
$$

{\rm (2)} If $p_1, p_2\in C_+(\overline\Omega), p_1(x)\le p_2(x)$
for any $x\in \Omega$, then there exists the continuous embedding
$L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$, whose norm
does not exceed $|\Omega|+1$.
\end{lemma}


\begin{lemma}[\cite{FZ}] \label{lem2}
If we denote
$$
\rho(u)=\int_\Omega |u|^{p(x)}\,dx, \quad \forall u\in
L^{p(x)}(\Omega),
$$
then
$$
\min\{\|u\|_{L^{p(x)}(\Omega)}^{p_-},\|u\|_{L^{p(x)}(\Omega)}^{p_+}\}\le \rho(u)\le
\max\{\|u\|_{L^{p(x)}(\Omega)}^{p_-},\|u\|_{L^{p(x)}(\Omega)}^{p_+}\}.
$$
\end{lemma}

\begin{lemma}[\cite{FZ}] \label{lem3}
$W^{k,p(x)}(\Omega)$ is a separable and reflexive Banach space.
\end{lemma}

\begin{lemma}[\cite{HHKV,KR}] \label{lem4}
Let  $p \in C_+(\overline\Omega)$ satisfy the
log-H\"{o}lder continuity condition {\rm \eqref{assume}}. Then, for
$u \in W_0^{1,p(\cdot)}(\Omega)$, the $p(\cdot)$-Poincar\'{e}
inequality
$$
\|u\|_{L^{p(x)}(\Omega)}\le C\|\nabla u\|_{L^{p(x)}(\Omega)}
$$
holds, where the positive constant $C$ depends on $p$, $N$ and $\Omega$.
\end{lemma}

\begin{lemma}[\cite{Di2, FZZ}] \label{lem6}
Let $\Omega\subset \mathbb{R}^N$ be an open, bounded set
with Lipschitz boundary and $p(x)\in C_+(\overline\Omega)$ with
$1<p_-\le p_+< N$ satisfy the log-H\"{o}lder continuity condition
{\rm \eqref{assume}}. If $q\in L^\infty(\Omega)$ with $q_->1$
satisfies
$$
q(x)\le p^*(x):=\frac{Np(x)}{N-p(x)},\quad\forall x\in \Omega,
$$
then we have
$$
W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)
$$
and the imbedding is compact if $\inf_{x\in
\Omega}(p^*(x)-q(x))>0$.
\end{lemma}

A variable exponent version of the relative $p(\cdot)$-capacity of the condenser
has been used in \cite{HHK}. This alternative capacity of a set is taken
relative to a surrounding open subset of $\mathbb{R}^N$. Suppose that
$p_+<\infty$ and $p(x)$ satisfies the log-H\"{o}lder continuity
condition \eqref{assume}. Let $K\subset \Omega$. The relative $p(\cdot)$-capacity of $K$ in $\Omega$ is the number
$$
\operatorname{cap}_{p(\cdot)}(K, \Omega)
=\inf\Big\{\int_\Omega |\nabla \varphi|^{p(x)}\,dx: \varphi\in C_0^\infty(\Omega)
\text{ and } \varphi\ge 1 \text{ in } K\Big\}.
$$
For an open set $U\subset \Omega$ we define
$$
\operatorname{cap}_{p(\cdot)}(U, \Omega)
=\sup\big\{\operatorname{cap}_{p(\cdot)}(K, \Omega): K\subset U \text{ compact}\big\}
$$
and for an arbitrary $E\subset \Omega$,
$$
\operatorname{cap}_{p(\cdot)}(E, \Omega)
=\inf\big\{\operatorname{cap}_{p(\cdot)}(U, \Omega): U\supset E \text{ open}\big\}.
$$
Then
$$
\operatorname{cap}_{p(\cdot)}(E, \Omega)
=\sup\big\{\operatorname{cap}_{p(\cdot)}(K, \Omega): K\subset E \text{ compact}\big\}
$$
for all Borel sets $E\subset \Omega$. The number
 $\operatorname{cap}_{p(\cdot)}(E, \Omega)$ is called the
variational $p(\cdot)$-capacity of $E$ relative to $\Omega$. We
usually call it simply the relative $p(\cdot)$-capacity of the pair. 
The relative $p(\cdot)$-capacity is an outer capacity.

We say that a function $f: \Omega \to \overline{\mathbb{R}}$ is
$p(\cdot)$-quasi continuous if for every $\varepsilon>0$ there
exists an open set $A\subset \Omega$ with $\operatorname{cap}_{p(\cdot)}(A,
\Omega)\le \varepsilon$, such that $f|_{\Omega\backslash A}$ is
continuous. Every $u\in W^{1,p(\cdot)}(\Omega)$ has a
$p(\cdot)$-quasi continuous representative (see \cite{BGO,HHK}), 
always denoted in this
paper by $\tilde{u}$, which is essentially unique.

Denote by $\mathcal {M}_b(\Omega)$ the space of all signed
measures on $\Omega$, i.e., the space of all $\sigma$-additive set
functions $\mu$ with values in $\mathbb{R}$ defined on the Borel
$\sigma$-algebra. If $\mu$ belongs to $\mathcal
{M}_b(\Omega)$, then $|\mu|$ (the total variation of $\mu$) is a
bounded positive measure on $\Omega$. We will denote by $\mathcal
{M}^{p(\cdot)}_b(\Omega)$ the space of all measures $\mu$ in
$\mathcal {M}_b(\Omega)$ such that $\mu(E)=0$ for every set $E$
satisfying $\operatorname{cap}_{p(\cdot)}(E, \Omega)=0$. Examples of
measures in $\mathcal {M}^{p(\cdot)}_b(\Omega)$ are the
$L^1(\Omega)$ functions, or the measures in
$W^{-1,p'(\cdot)}(\Omega)$.

Next we have a decomposition of a measure in $\mathcal{M}^{p(\cdot)}_b(\Omega)$.

\begin{proposition} \label{prop4}
Assume that $p(x)$ satisfies the log-H\"{o}lder condition \eqref{assume}
with $1<p_-\le p_+<+\infty$. Let $\mu$ be an element of 
$\mathcal {M}_b(\Omega)$.
Then $\mu\in L^1(\Omega)+W^{-1,p'(\cdot)}(\Omega)$ if and only if
$\mu\in \mathcal {M}^{p(\cdot)}_b(\Omega)$. Thus, if $\mu\in
\mathcal {M}^{p(\cdot)}_b(\Omega)$, there exist $f$ in $L^1(\Omega)$
and $F$ in $(L^{p'(\cdot)}(\Omega))^N$, such that
$$
\mu=f-\operatorname{div}F,
$$
in the sense of distributions.
\end{proposition}

\begin{proof}  \emph{Necessity.}
 If $\mu$ belongs to
$L^1(\Omega)+W^{-1,p'(\cdot)}(\Omega)$, then there exist $f\in
L^1(\Omega)$ and $F\in L^{p'(\cdot)}(\Omega)$ such that $\mu=f-\operatorname{div}
F$. We just need to show that $\mu(E)=0$ for every set $E\subset \Omega$
such that $\operatorname{cap}_{p(\cdot)}(E, \Omega)=0$. It is easy to see
that $\mu \in \mathcal {M}_b(\Omega)$. From the definition of $p(\cdot)$-capacity and the similar arguments
as in Lemma 2.4 of \cite{M}, there is a Borel set $E_0\subset \Omega$ such that $E\subset E_0$ and
$\operatorname{cap}_{p(\cdot)}(E_0,\Omega)=0$. Let $K\subset E_0$ be compact and $\Omega' \subset \Omega$ an open
set containing $K$. Then there is a sequence $(\varphi_j)\subset C_0^\infty(\Omega')$ such that $0\le \varphi_j\le 1$,
$\varphi_j=1$ in $K$ and $\int_{\Omega'}|\nabla \varphi_j|^{p(x)}dx\to 0$ as $j\to \infty$. Then we have
\[
|\mu(K)|\le \big|\int_{\Omega'} \varphi_j\,d\mu\big|
 \le \big|\int_{\Omega'} f\varphi_j\,dx+\int_{\Omega'} F\cdot
\nabla\varphi_j\,dx\big|.
\]
Choosing the regular functions $\{f_n\}$ such that 
$\|f_n-f\|_{L^1(\Omega)}\to 0$ as $n\to \infty$ and applying
Lemmas \ref{lem1}, \ref{lem2} and \ref{lem4} yield that
\begin{align*}
|\mu(K)| &\le \int_{\Omega'} |f_n-f|\cdot |\varphi_j|\,dx+\int_{\Omega'}
|f_n|\cdot |\varphi_j|\,dx+\int_{\Omega'} |F|\cdot |\nabla \varphi_j|\,dx\\
&\le
\|\varphi_j\|_{L^\infty(\Omega')}\|f_n-f\|_{L^1(\Omega')}+2\|f_n\|_{L^{p'(x)}(\Omega')}
\|\varphi_j\|_{L^{p(x)}(\Omega')}\\
&\quad+2\|F\|_{L^{p'(x)}(\Omega')}\|\nabla\varphi_j\|_{L^{p(x)}(\Omega')}\\
&\le
\|\varphi_j\|_{L^\infty(\Omega')}\|f_n-f\|_{L^1(\Omega')}+C\|f_n\|_{L^{p'(x)}(\Omega')}
\|\nabla\varphi_j\|_{L^{p(x)}(\Omega')}\\
&\quad+2\|F\|_{L^{p'(x)}(\Omega')}\|\nabla\varphi_j\|_{L^{p(x)}(\Omega')}\\
&\le
\|\varphi_j\|_{L^\infty(\Omega')}\|f_n-f\|_{L^1(\Omega')}+C\|f_n\|_{L^{p'(x)}(\Omega')}
\Big(\int_{\Omega'} |\nabla \varphi_j|^{p(x)}\,dx\Big)^{\gamma}\\
&\quad+2\|F\|_{L^{p'(x)}(\Omega')}\Big(\int_{\Omega'} |\nabla
\varphi_j|^{p(x)}\,dx\Big)^{\gamma},
\end{align*}
where
\[
\gamma=\begin{cases} 
1/p_- & \text{if } \|\nabla \varphi_j\|_{L^{p(x)}(\Omega')}\ge 1,\\
1/p_+ & \text{if } \|\nabla \varphi_j\|_{L^{p(x)}(\Omega')}\le 1.
\end{cases} 
\]
It follows that for all compact $K\subset E_0$,
$$
|\mu(K)| \le C\|f_n-f\|_{L^1(\Omega')} \quad \text{as} \quad j\to \infty,
$$
where $C$ is a positive constant that does not depend on $n$. Moreover,
 it implies that $\mu(K)=0$ as $n\to \infty$, and then 
$\mu(E)\le \mu(E_0)=\sup\{\mu(K): K\subset E_0 \text{ compact}\}=0$ 
by the regularity of $\mu$.

\emph{Sufficiency.}
 Motivated by the ideas developed in \cite{BGO,DM,DPP} with constant exponents,
 we sketch the proof. In the following we assume that $\mu$ is positive. 
(If not, we write $\mu=\mu^+-\mu^-$.)

\textbf{Step $1$.} First we prove that every measure $\mu$ in
 $\mathcal {M}^{p(\cdot)}_b(\Omega)$ can be decomposed as 
$\mu=f\gamma^{\rm meas}$, i.e., $d\mu=fd\gamma^{\rm meas}$, with $f$ a 
positive Borel measurable function in $L^1(\Omega, \gamma^{\rm meas})$ and
$\gamma^{\rm meas}$ a positive measure in
$W^{-1,p'(\cdot)}(\Omega)$. Indeed, for any $u\in
W_0^{1,p(\cdot)}(\Omega)$, let $\tilde{u}$ be the $p(\cdot)$-quasi
continuous representative of $u$. Since $\tilde{u}$ is uniquely
defined up to sets of zero $p(\cdot)$-capacity, we can define the
functional $F: W_0^{1,p(\cdot)}(\Omega)\to [0, +\infty]$ by
$$
F(u)=\int_{\Omega} \max\{\tilde{u},0\}\,d\mu.
$$
Clearly, $F$ is convex and lower semi-continuous on
$W_0^{1,p(\cdot)}(\Omega)$. Since $W^{1,p(\cdot)}(\Omega)$ is
separable from Lemma \ref{lem3}, the function $F$ is the supremum of
a countable family of continuous affine functions. Therefore, there
exist a sequence $\{\lambda_n\}$ in $W^{-1, p'(\cdot)}(\Omega)$ and
a sequence $\{a_n\}$ in $\mathbb{R}$ such that
$$
F(u)=\sup_{n\in \mathbb N}\{\langle\lambda_n,
u\rangle+a_n\}
$$
for every $u\in W_0^{1,p(\cdot)}(\Omega)$. Since, for any positive
$t$, $tF(u)=F(tu)\ge t\langle\lambda_n, u\rangle+a_n$ for every $n$,
dividing by $t$ and let $t\to +\infty$, we get $F(u)\ge
\langle\lambda_n, u\rangle$ for all $u\in W_0^{1,p(\cdot)}(\Omega)$.
For $u=0$, we deduce that $a_n\le 0$. Thus
\begin{equation}
\label{e2-1}
 F(u)\ge \sup_n\langle\lambda_n, u\rangle\ge
\sup_n\{\langle\lambda_n, u\rangle+a_n\}=F(u),
\end{equation}
which implies that
\begin{equation}
\label{e2-2} F(u)=\sup_{n\in \mathbb N}\langle\lambda_n,
u\rangle.
\end{equation}
In view of \eqref{e2-2} and the definition of $F$, for all $\varphi\in
C_0^\infty(\Omega)$, we have
\begin{equation} \label{e2-3}
\langle\lambda_n, \varphi\rangle\le
\sup_n\langle\lambda_n, \varphi\rangle=F(\varphi)=\int_\Omega
\varphi^+\,d\mu\le \|\mu\|_{\mathcal
{M}_b(\Omega)}\|\varphi\|_{L^\infty(\Omega)}.
\end{equation}
Thus, applying this inequality to $\varphi$ and $-\varphi$, we obtain
$$
|\langle\lambda_n, \varphi\rangle|\le \|\mu\|_{\mathcal
{M}_b(\Omega)}\|\varphi\|_{L^\infty(\Omega)},
$$
which implies that $\lambda_n\in W^{-1, p'(\cdot)}(\Omega)\cap
\mathcal {M}_b(\Omega)$. Moreover, since $F(-\varphi)=0$ for any
nonnegative $\varphi\in C_0^\infty(\Omega)$, we have $\langle
\lambda_n,\varphi\rangle\ge 0$. By the Riesz representation theorem
there exists a nonnegative measure on $\Omega$, which we denote by
$\lambda_n^{\rm meas}$, such that
$$
\langle \lambda_n,\varphi\rangle=\int_\Omega
\varphi\,d\lambda_n^{\rm meas}, \quad\text{for all such } \varphi,
$$
which implies $\lambda_n^{\rm meas}\in \mathcal {M}_b^+(\Omega)$
(that is to say $\lambda_n\in W^{-1,p'(\cdot)}(\Omega)\cap \mathcal
{M}_b^+(\Omega)$). Using again \eqref{e2-3} to any nonnegative
$\varphi\in C_0^\infty(\Omega)$, we obtain
\begin{equation} \label{e2-4}
\lambda_n^{\rm meas}\le \mu, \quad \|\lambda_n^{\rm
meas}\|_{\mathcal {M}_b(\Omega)}\le \|\mu\|_{\mathcal
{M}_b(\Omega)}.
\end{equation}

Define
\begin{equation}\label{e2-5}
\gamma=\sum_{n=1}^\infty
\frac{\lambda_n}{2^n(\|\lambda_n\|_{W^{-1,p'(\cdot)}(\Omega)}+1)}.
\end{equation}
It is obvious that the series is absolutely convergent in
$W^{-1,p'(\cdot)}(\Omega)$. Then we have, for all $\varphi\in
C_0^\infty(\Omega)$,
\begin{align*}
|\langle \gamma,\varphi\rangle|
&=\big|\sum_{n=1}^\infty
\frac{\langle\lambda_n,\varphi\rangle}{2^n(\|\lambda_n\|_{W^{-1,p'(\cdot)}
(\Omega)}+1)}\big|\\
&\le \sum_{n=1}^\infty \frac{\|\lambda_n^{\rm meas}\|_{\mathcal
{M}_b(\Omega)}\|\varphi\|_{L^\infty(\Omega)}}{2^n}\\
&\le \|\mu\|_{\mathcal {M}_b(\Omega)}\|\varphi\|_{L^\infty(\Omega)},
\end{align*}
and $\gamma\in W^{-1,p'(\cdot)}(\Omega)\cap \mathcal {M}_b(\Omega)$.
Since the series $\sum_{n=1}^\infty\frac{\lambda_n^{\rm
meas}}{2^n(\|\lambda_n\|_{W^{-1,p'(\cdot)}(\Omega)}+1)}$ strongly
converges in $\mathcal {M}_b(\Omega)$. Applying \eqref{e2-5} to
functions of $C_0^\infty(\Omega)$, we can see that
$$
\gamma^{\rm meas}=\sum_{n=1}^\infty \frac{\lambda_n^{\rm
meas}}{2^n(\|\lambda_n\|_{W^{-1,p'(\cdot)}(\Omega)}+1)}.
$$
In particular, $\gamma^{\rm meas}$ is a nonnegative measure (each
$\lambda_n^{\rm meas}$ is nonnegative).

Since $\lambda_n^{\rm meas}\ll \gamma^{\rm meas}$, there exists a
nonnegative function $f_n\in L^1(\Omega, d\gamma^{\rm meas})$ such
that $\lambda_n^{\rm meas}=f_n \gamma^{\rm meas}$. Thus \eqref{e2-2}
implies
\begin{equation}\label{e2-6}
\int_\Omega \varphi\,d\mu=\sup_n\int_\Omega f_n\varphi\,
d\gamma^{\rm meas},
\end{equation}
for any nonnegative $\varphi\in C_0^\infty(\Omega)$. We also have,
by \eqref{e2-4}, $f_n\gamma^{\rm meas}\le \mu$, that is
\begin{equation}
\label{e2-8} \int_B f_n\,d\gamma^{\rm meas}\le \mu(B),
\end{equation}
for any Borelian subset $B\subset \Omega$ and every $n$.

Denote
\[
B_s=\big\{x\in B: f_s(x)=\max\{f_1(x),\dots,f_k(x)\} \text{ and }
f_s(x)>f_1(x),\dots,f_{s-1}(x)\big\}.
\]
It is obvious that $B_i$  $(i=1,\dots,k)$ are disjoint and
$B=\cup_{s=1}^k B_s$.
Then by \eqref{e2-8} we have
$$
\int_{B_s} f_s\,d\gamma^{\rm meas}\le \mu(B_s);
$$
that is,
$$
\int_{B_s} \sup\{f_1,\dots,f_k\}\,d\gamma^{\rm meas}\le \mu(B_s).
$$
Summing up the above inequalities for $s=1,\dots,k$, we deduce that
$$
\int_B \sup\{f_1,\dots,f_k\}\,d\gamma^{\rm meas}\le \mu(B),
$$
for any Borelian subset $B\subset \Omega$ and any $k\ge 1$. Letting
$k\to \infty$, we obtain from the monotone convergence theorem that
$$
\int_B f\,d\gamma^{\rm meas}\le \mu(B),
$$
where $f=\sup_n f_n$. Then from \eqref{e2-6} we conclude that
\begin{align*}
\int_\Omega \varphi\,d\mu
&=\sup_n\int_\Omega
f_n\varphi\,d\gamma^{\rm meas}\le \sup_n\int_\Omega f \varphi\,d\gamma^{\rm meas}\\
&=\int_\Omega f \varphi\,d\gamma^{\rm meas}\le \int_\Omega
\varphi\,d\mu,
\end{align*}
for any nonnegative $\varphi\in C_0^\infty(\Omega)$, which yields
that
$$
\mu=f\gamma^{\rm meas}.
$$
Since $\mu(\Omega)<+\infty$, it follows that $f\in L^1(\Omega,
d\gamma^{\rm meas})$.
\smallskip

\textbf{Step $2$.} 
Let $K_n$ be an increasing sequence of compact sets
contained in $\Omega$ such that $\cup_{n=1}^{+\infty}K_n=\Omega$. Denote
$\mu_n^{(1)}=T_n(f\chi_{K_n})\gamma^{\rm meas}$. 
It is obvious that $\{\mu_n^{(1)}\}$ is an increasing sequence of positive measure
in $W^{-1,p'(\cdot)}(\Omega)$ with compact support in $\Omega$. Set
$\mu_0=\mu_0^{(1)}$ and $\mu_n=\mu_n^{(1)}-\mu_{n-1}^{(1)}$. Then
$\mu=\sum_{n=1}^{+\infty}\mu_n$, and the series converges strongly
in $\mathcal {M}_b(\Omega)$. Since $\mu_n\ge 0$ and $\|\mu_n\|_{\mathcal
{M}_b(\Omega)}=\mu_n(\Omega)$, we know that
 $\sum_{n=1}^{+\infty}\|\mu_n\|_{\mathcal {M}_b(\Omega)}<\infty$.
\smallskip

\textbf{Step $3$.}  
Let $\rho\ge 0$ be a function in $C_0^\infty(\mathbb{R}^N)$ with
$\int_{\mathbb{R}^N}\rho(x)\,dx=1$. Let $\{\rho_n\}$ be a sequence of mollifiers
associated to $\rho$; i.e., $\rho_n(x)=n^N \rho(nx)$ for every $x\in
\mathbb{R}^N$. For $n\in \mathbb N$, if $\mu_n$ is the measure
defined in Step $2$, the log-H\"{o}lder continuity condition \eqref{assume}
implies that $\{\mu_n\ast \rho_m\}$ converges to
$\mu_n$ in $W^{-1,p'(\cdot)}(\Omega)$ as $m$ tends to infinity. By
the properties of $\mu_n$ and $\rho_m$, $\mu_n\ast \rho_m$ belongs
to $C_0^\infty(\Omega)$ if $m$ is large enough.

Choose $m=m_n$ such that $\mu_n\ast \rho_{m_n}$ belongs to
$C_0^\infty(\Omega)$ and
$\|\mu_n\ast\rho_{m_n}-\mu_n\|_{W^{-1,p'(\cdot)}(\Omega)}\le
2^{-n}$. Then $\mu_n=f_n+g_n$, where $f_n=\mu_n\ast \rho_{m_n}$ and
$g_n=\mu_n-\mu_n\ast \rho_{m_n}$. The choice of $m_n$ implies that the
series $\sum_{n=1}^{+\infty}g_n$ converges in
$W^{-1,p'(\cdot)}(\Omega)$ and $g=\sum_{n=1}^{+\infty}g_n$
belongs to $W^{-1,p'(\cdot)}(\Omega)$. Since
$\|f_n\|_{L^1(\Omega)}=\|\mu_n\ast \rho_{m_n}\|_{L^1(\Omega)}\le
\|\mu_n\|_{\mathcal {M}_b(\Omega)}$, by Step $2$ the series
$\sum_{n=1}^{+\infty}f_n$ is absolutely convergent in $L^1(\Omega)$,
and $f_0=\sum_{n=1}^{+\infty}f_n$ belongs to $L^1(\Omega)$. Therefore,
the three series $\sum_{n=1}^{+\infty}\mu_n, \sum_{n=1}^{+\infty}g_n$ 
and $\sum_{n=1}^{+\infty}f_n$ converge in
the sense of distributions. Then $\mu=f_0+g$. This completes the proof.
\end{proof}


\begin{remark} \rm
From Proposition \ref{prop4}, we can conclude that 
$\mu\in  {M}^{p(\cdot)}_b(\Omega)$
is a signed measure in $L^1(\Omega)+W^{-1,p'(\cdot)}(\Omega)$; i.e.,
\[
\mu=f-\operatorname{div}F \quad\text{in the sense of distributions},
\]
where $f\in L^1(\Omega)$ and $F\in (L^{p'(\cdot)}(\Omega))^N$. 
Therefore, the equality \eqref{edef-inequality} can be written as
\begin{equation} \label{e1-4}
\begin{aligned}
&\int_\Omega |\nabla u|^{p(x)-2}\nabla u\cdot \nabla
T_k(u-\phi)\,dx   \\
&= \int_\Omega f T_k(u-\phi)\,dx+\int_\Omega F\cdot \nabla
T_k(u-\phi)\,dx,
\end{aligned}
\end{equation}
for all $\phi\in W^{1,p(x)}_0(\Omega)\cap L^\infty(\Omega)$.
\end{remark}

Based on the decomposition of a measure in $\mathcal{M}^{p(\cdot)}_b(\Omega)$,
 we have the following result, whose proof can be found in \cite{ZZ}.

\begin{theorem} 
Assume that $p(x)$ satisfies the log-H\"{o}lder condition \eqref{assume}
and $\mu\in  {M}^{p(\cdot)}_b(\Omega)$. Then
there exists a unique entropy solution $u\in\mathcal {T}_0^{1,p(\cdot)}(\Omega)$ 
for problem \eqref{eprob1}.
\end{theorem}

\section{Weighted case}

In this section, we are ready to prove the existence of entropy solutions 
for weighted $p(x)$-Laplace problem \eqref{emain}.

\subsection{Preliminaries}
Let $w$ be a weight function satisfying that
\begin{itemize}
\item [(W1)] $w\in L_{\rm loc}^1(\Omega)$ and $w^{-1/(p(x)-1)}\in L_{\rm
loc}^1(\Omega)$;

\item [(W2)]$w^{-s(x)}\in L^1(\Omega)$ with $s(x)\in\big(\frac{N}{p(x)},\infty\big)
\cap [\frac{1}{p(x)-1},\infty)$.
\end{itemize}

\begin{lemma}[\cite{HD,KWZ}]\label{lem3-2}
If we denote
$$
\rho(u)=\int_\Omega w(x)|u|^{p(x)}\,dx, \quad \forall u\in
L^{p(x)}(\Omega,w),
$$
then
$$
\min\{\|u\|_{L^{p(x)}(\Omega,w)}^{p_-},\|u\|_{L^{p(x)}(\Omega,w)}^{p_+}\}
\le \rho(u)\le
\max\{\|u\|_{L^{p(x)}(\Omega,w)}^{p_-},\|u\|_{L^{p(x)}(\Omega,w)}^{p_+}\}.
$$
\end{lemma}


\begin{lemma}[\cite{KWZ}]\label{lem3-3}
If {\rm (W1)} holds, $W^{1,p(x)}(\Omega,w)$ is a separable and reflexive
Banach space.
\end{lemma}

For $p, s\in C_+(\overline{\Omega})$, set
$$
p_{s}(x):=\frac{p(x)s(x)}{1+s(x)}<p(x),
$$ 
where $s(x)$ is given in (W2). Assume that we fix the variable exponent 
restrictions
\begin{equation}
p^*_{s}(x):= \begin{cases}
\frac{p(x)s(x)N}{(s(x)+1)N-p(x)s(x)} &\text{if } N>p_{s}(x),\\
\text{arbitrary} &\text{if } N\le p_{s}(x),
\end{cases}
\end{equation}
for almost all $x\in\Omega$.

Next we state a continuous imbedding theorem for the
weighted variable exponent Sobolev space.

\begin{lemma}[\cite{KWZ}]\label{lem3-4}
Let $p, s\in C_+(\overline{\Omega})$ and let {\rm (W1)} and
{\rm (W2)} be satisfied. Then we have the continuous imbedding
$$
W^{1,p(x)}(\Omega,w)\hookrightarrow L^{r(x)}(\Omega)
$$
provided that $r\in C_+(\overline{\Omega})$ and $r(x)\le p_s^*(x)$ 
for all $x\in \Omega$ and the embedding is compact if 
$\inf_{x\in\Omega}(p_s^*(x)-r(x))>0$.
\end{lemma}

We conclude this subsection by proving {\it a priori} estimate for entropy 
solutions of problem \eqref{emain}, which plays a key role in proving 
our main result.

\begin{proposition} \label{prop2}
If $u$ is an entropy solution of problem \eqref{emain}, then there exists a positive
constant $C$ such that for all $k>1$,
$$
\operatorname{meas}\{|u|>k\}\le \frac{C(M+1)^{\frac{(p_s^*)_-}{p_-}}}{k^{(p_s^*)_
-(1-\frac{1}{p_-})}},
$$
where
$$
M=\|f\|_{L^1(\Omega)}, \quad  (p^*_{s})_-:=\frac{p_-s_-N}{(s_-+1)N-p_-s_-}.
$$
\end{proposition}

\begin{proof} 
Choosing $\phi=0$ in the entropy equality
\eqref{edef-inequality}, we obtain
\begin{align*}
\int_\Omega w(x)|\nabla T_k(u)|^{p(x)}\,dx=\int_{\{|u|\le k\}} w(x)|\nabla
u|^{p(x)}\,dx\le k\|f\|_{L^1(\Omega)},
\end{align*}
which implies that for all $k>1$,
\begin{equation} \label{e2-1b}
\frac 1k \int_\Omega w(x)|\nabla T_k(u)|^{p(x)}\,dx\le M,
\end{equation}
where $M=\|f\|_{L^1(\Omega)}$.

Recalling Sobolev embedding theorem in Lemma \ref{lem3-4}, we have the 
following continuous embedding
$$
W_0^{1,p(x)}(\Omega,w)\hookrightarrow L^{p_s^*(x)}(\Omega)
\hookrightarrow L^{(p_s^*)_-}(\Omega),
$$
where $p^*_{s}(x):=\frac{p(x)s(x)N}{(s(x)+1)N-p(x)s(x)}$ and 
$(p^*_{s})_-:=\frac{p_-s_-N}{(s_-+1)N-p_-s_-}$.
It follows from Lemma \ref{lem3-2} and \eqref{e2-1} that for every
$k>1$,
\begin{align*}
\|T_k(u)\|_{L^{(p_s^*)_-}(\Omega)}
&\le C\|\nabla T_k(u)\|_{L^{p(x)}(\Omega,w)}\\
&\le C\Big(\int_\Omega w(x)|\nabla T_k(u)|^{p(x)}\,dx\Big)^{\beta}\le
C(Mk)^{\beta},
\end{align*}
where
\[
\beta=\begin{cases} \frac{1}{p_-} 
& \text{if } \|\nabla T_k(u)\|_{L^{p(x)}(\Omega,w)}\ge 1,\\
\frac{1}{p_+} & \text{if }\|\nabla T_k(u)\|_{L^{p(x)}(\Omega,w)}\le 1.
\end{cases}
\]
Noting that $\{|u|\ge k\}=\{|T_k(u)|\ge k\}$, we have
\[
\operatorname{meas}\{|u|>k\}
\le \Big(\frac{\|T_k(u)\|_{L^{(p_s^*)_-}(\Omega)}}{k}\Big)^{(p_s^*)_-}
\le \frac{CM^{\beta (p_s^*)_-}}{k^{(p_s^*)_-(1-\beta)}}
\le
\frac{C(M+1)^{\frac{(p_s^*)_-}{p_-}}}{k^{(p_s^*)_-(1-\frac{1}{p_-})}}.
\]
This completes the proof.
\end{proof}

\subsection{Main result}

\begin{theorem}
Let {\rm (W1)} and {\rm (W2)} be satisfied. Then there exists an entropy 
solution for problem \eqref{emain}.
\end{theorem}

\begin{proof}
We first introduce the approximation problems.
Find a sequence of $C^\infty_0(\Omega)$
functions $\{f_n\}$ strongly converging to $f$ in $L^1(\Omega)$
such that
\begin{align}\label{eassume}
\|f_n\|_{L^1(\Omega)}\le C\big(\|f\|_{L^1(\Omega)}+1\big).
\end{align}
Then we consider  approximate problems of \eqref{emain}
\begin{equation}\label{appro}
\begin{gathered}
-\operatorname{div} \big(w(x)|\nabla u_n|^{p(x)-2}\nabla u_n\big)
=f_n \quad \text{in }  \Omega,\\
u_n=0 \quad \text{on }  \partial\Omega.
\end{gathered}
\end{equation}
Then from the result in \cite{F}, we can easily find a unique weak solution
$u_n\in W_0^{1,p(\cdot)}(\Omega,w)$ of problem \eqref{appro}, which is 
obviously an entropy solution, satisfying that for all
$\phi\in W_0^{1,p(x)}(\Omega,w)\cap L^\infty(\Omega)$,
\[
\int_\Omega w(x)|\nabla u_n|^{p(x)-2}\nabla u_n\cdot \nabla
T_k(u_n-\phi)\,dx=\int_\Omega f_n T_k(u_n-\phi)\,dx.
\]
Following the same arguments as in Proposition \ref{prop2} and \eqref{assume},
 we have
\begin{equation} \label{e3-3}
\int_{\Omega} w(x)|\nabla T_k(u_n)|^{p(x)}\,dx\le Ck(\|f\|_{L^1(\Omega)}+1).
\end{equation}
Our aim is to prove that a subsequence of these approximate solutions $\{u_n\}$
converges to a measurable function $u$, which is an entropy solution
of problem \eqref{emain}. We will divide the proof into several
steps.
\smallskip

\textbf{Step $1$.} 
We shall prove the convergence in measure of $\{u_n\}$ and we shall find a 
subsequence which is almost everywhere convergent in $\Omega$.
For every fixed $\epsilon>0$, and every positive integer $k$, we
know that
$$
\{|u_n-u_m|>\epsilon\}\subset \{|u_n|>k\} \cup \{|u_m|>k\}
\cup\{|T_k(u_n)-T_k(u_m)|>\epsilon\}.
$$
Using Sobolev embedding theorem in Lemma \ref{lem3-4}, we find that 
$W^{1,p(x)}(\Omega,w)$ can embed into $L^q(\Omega)$ with $q<(p_s^*)_-$ compactly. 
Then we know $\{T_ku_n\}$ is convergent in $L^q(\Omega)$ with $q<(p_s^*)_-$. 
It follows from Proposition \ref{prop2} that
$$
\limsup_{n,m\to \infty}\operatorname{meas}\{|u_n-u_m|>\epsilon\}\le
Ck^{-\alpha},
$$
where $\alpha=(p_s^*)_-(1-\frac{1}{p_-})>0$ and the constant $C$ depends 
on $p(\cdot), s(\cdot)$ and  $\|f\|_{L^1(\Omega)}$.

Because of the arbitrariness of $k$, we prove that
$$
\limsup_{n,m\to \infty}\operatorname{meas}\{|u_n-u_m|>\epsilon\}=0,
$$
which implies the convergence in measure of $\{u_n\}$,  and then we
find an a.e. convergent  subsequence  (still denoted by $\{u_n\}$)
in $\Omega$ such that
\begin{equation}\label{e3-4}
u_n\to u \quad \text{a.e. in } \Omega.
\end{equation}
\smallskip

\textbf{Step $2$.} We shall prove that
\begin{equation} \label{e3-5}
\nabla T_k(u_n) \to \nabla T_k(u) \quad \text{strongly in } W_0^{1,p(x)}(\Omega,w),
\end{equation}
for every $k>0$.
Let $h>k$. We choose
$$
w_n=T_{2k}\big(u_n-T_h(u_n)+T_k(u_n)-T_k(u)\big)
$$
as a test function in \eqref{appro}. If we set $M=4k+h$, then it is
easy to see that $\nabla w_n=0$ where $\{|u_n|>M\}$. Therefore, we may
write the weak form  of \eqref{appro} as
$$
\int_\Omega w(x)|\nabla T_M(u_n)|^{p(x)-2}\nabla T_M(u_n)\cdot \nabla
w_n\,dx=\int_\Omega f_n w_n\,dx.
$$
Splitting the integral in the left-hand side on the sets where
$\{|u_n|\le k\}$ and where $\{|u_n|>k\}$ and discarding some nonnegative
terms, we find
\begin{align*}
&\int_\Omega w(x)|\nabla T_M(u_n)|^{p(x)-2}\nabla T_M(u_n)\cdot \nabla
T_{2k}(u_n-T_h(u_n)+T_k(u_n)-T_k(u))\,dx
\\
&\ge \int_\Omega  w(x)|\nabla T_k(u_n)|^{p(x)-2}\nabla T_k(u_n)\cdot
\nabla (T_k(u_n)-T_k(u))\,dx
\\
&\quad-\int_{\{|u_n|>k\}} w(x)\big||\nabla T_M(u_n)|^{p(x)-2}\nabla
T_M(u_n)\big||\nabla T_k(u)|\,dx.
\end{align*}
It follows from the above inequality  that
\begin{equation} \label{e3-7}
\begin{aligned}
&\int_\Omega w(x)\left(|\nabla T_k(u_n)|^{p(x)-2}\nabla
T_k(u_n)-|\nabla T_k(u)|^{p(x)-2}\nabla T_k(u)\right)\\
&\quad \cdot \nabla(T_k(u_n)-T_k(u))\,dx  \\
&\le \int_{\{|u_n|>k\}} w(x)\big||\nabla T_M(u_n)|^{p(x)-2}\nabla
T_M(u_n)\big||\nabla T_k(u)|\,dx \\
&\quad+\int_\Omega f_n T_{2k}(u_n-T_h(u_n)+T_k(u_n)-T_k(u))\,dx\\
&\quad-\int_\Omega w(x)|\nabla T_k(u)|^{p(x)-2}\nabla T_k(u)\cdot
\nabla(T_k(u_n)-T_k(u))\,dx  \\
&:=I_1+I_2+I_3.
\end{aligned}
\end{equation}
Using the properties of $L^{p(x)}(\Omega,w)$ and the similar estimates
as in \cite{C}, we can show the limits of $I_1$, $I_2$ and $I_3$ are zeros
 when $n$, and then $h$ tend to infinity, respectively.

Therefore, passing to the limits in \eqref{e3-7} as $n$, and then $h$
tend to infinity, we deduce that
$$
\lim_{n\to +\infty}E(n)=0,
$$
where
\begin{align*}
E(n)&=\int_\Omega w(x)(|\nabla T_k(u_n)|^{p(x)-2}\nabla T_k(u_n)\\
&\quad -|\nabla T_k(u)|^{p(x)-2}\nabla T_k(u))
 \nabla(T_k(u_n)-T_k(u))\,dx.
\end{align*}
Applying \cite[Lemma 3.1]{C}, we conclude that
$$
T_k(u_n)\to T_k(u) \quad\text{strongly in } W_0^{1,p(x)}(\Omega,w)
$$
for every $k>0$, which also  implies that
\begin{equation*}
|\nabla T_k(u_n)|^{p(x)-2}\nabla T_k(u_n)\to |\nabla
T_k(u)|^{p(x)-2}\nabla T_k(u) \quad \text{strongly in }
(L^{p'(\cdot)}(\Omega,w))^N.
\end{equation*}
\smallskip


\textbf{Step $3$.} We shall prove that $u$ is an entropy solution.
Set $L=k+\|\phi\|_{L^\infty(\Omega)}$. Observe that
\begin{align*}
&\int_\Omega w(x)|\nabla u_n|^{p(x)-2}\nabla u_n\cdot \nabla
T_k(u_n-\phi)\,dx\\
&=\int_\Omega |\nabla T_L(u_n)|^{p(x)-2}\nabla
T_L(u_n)\cdot \nabla T_k(u_n-\phi)\,dx.
\end{align*}
Then we have
\[
\int_\Omega w(x)|\nabla T_L(u_n)|^{p(x)-2}\nabla T_L(u_n)\cdot \nabla
T_k(u_n-\phi)\,dx
=\int_\Omega f_n T_k(u_n-\phi)\,dx.
\]
Using \eqref{e3-4} and \eqref{e3-5}, we can pass to the limits as $n$
tends to infinity to conclude that
\[ 
\int_\Omega w(x)|\nabla u|^{p(x)-2}\nabla u\cdot \nabla T_k(u-\phi)\,dx
= \int_\Omega f T_k(u-\phi)\,dx,
\]
for every $k>0$ and every $\phi\in W_0^{1,p(x)}(\Omega,w) \cap L^\infty(\Omega)$. 
This finishes the proof.
\end{proof}

\subsection*{Acknowledgments}
The author wishes to thank the anonymous reviewer for offering valuable 
suggestions to improve this article. The author would like to thank 
Professor Shulin Zhou for the helpful conversations. 
This work was supported by the NSFC (Nos. 11201098, 11301113), 
Research Fund for the Doctoral Program of Higher Education of China 
(No. 20122302120064), the Fundamental Research Funds for the Central 
Universities (No. HIT. NSRIF. 2013080), the PIRS of HIT A201406,
and the China Postdoctoral Science Foundation (No. 2012M510085).


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