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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 68, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/68\hfil Existence and regularity of entropy solutions]
{Existence and regularity of entropy solutions for strongly
  nonlinear $p(x)$-elliptic equations}

\author[E. Azroul, H. Hjiaj, A. Touzani \hfil EJDE-2013/68\hfilneg]
{Elhoussine Azroul, Hassane Hjiaj, Abdelfattah Touzani}  % in alphabetical order

\address{Elhoussine Azroul \newline
University of Fez, Faculty of Sciences Dhar El Mahraz,
Laboratory LAMA, Department of Mathematics,
B.P. 1796 Atlas  Fez, Morocco}
\email{azroul\_elhoussine@yahoo.fr}

\address{Hassane Hjiaj \newline
University of Fez, Faculty of Sciences Dhar El Mahraz,
Laboratory LAMA, Department of Mathematics,
B.P. 1796 Atlas  Fez, Morocco}
\email{hjiajhassane@yahoo.fr}

\address{Abdelfattah Touzani \newline
University of Fez, Faculty of Sciences Dhar El Mahraz,
Laboratory LAMA, Department of Mathematics,
B.P. 1796 Atlas  Fez, Morocco}
\email{atouzani07@gmail.com}

\thanks{Submitted May 22, 2012. Published March 8, 2013.}
\subjclass[2000]{35J20, 35J25, 35J60}
\keywords{Sobolev spaces with variable exponents; entropy solutions;
\hfill\break\indent strongly nonlinear elliptic equations; boundary value problems}

\begin{abstract}
 This article is devoted to study the existence of solutions for
 the  strongly nonlinear $p(x)$-elliptic problem
 \begin{gather*}
 - \operatorname{div}  a(x,u,\nabla u) + g(x,u,\nabla u)
 = f- \operatorname{div}  \phi(u) \quad \text{in } \Omega, \\
  u  =   0 \quad \text{on }  \partial\Omega,
 \end{gather*}
 with $ f\in L^1(\Omega) $ and $ \phi \in  C^{0}(\mathbb{R}^{N})$,
 also we will give some  regularity results for these solutions.
\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}

Let $ \Omega  $ be a bounded open subset of $\mathbb{R}^{N}$  with $N\geq 2$.
For $ 2 - \frac{1}{N} < p < N$,  Boccardo and  Gallou\"{e}t \cite{bocc3}
studied the problem
\begin{gather*}
Au  = f  \quad \text{in }  \Omega, \\
   u  =   0\quad \text{on }  \partial\Omega,
\end{gather*}
where $ Au = - \operatorname{div}   a(x,u,\nabla u) $
is a Leray-Lions operator from $ W_0^{1,p}(\Omega) $ into its dual,
and $ f $ is a bounded Radon measure on $ \Omega$.
They  proved the existence of solutions $ u\in W_0^{1,q}(\Omega) $
for all $ 1 < q < \bar{q} = \frac{N(p-1)}{N-1}$.
 Moreover, they showed the critical regularity
 $ u\in W_0^{1,\overline{q}}(\Omega) $  under the assumption
$ f\log(1 + |f|) \in L^1(\Omega)$.
 Boccardo \cite{bocc1}  studied the existence of entropy solutions
for the problem
\begin{equation}
\begin{gathered}
- \operatorname{div}   a(x,u,\nabla u)  = f- \operatorname{div}  \phi(u) \quad
\text{in }  \Omega, \\
   u  =   0\quad \text{on }  \partial\Omega,
\end{gathered}\label{appp}
\end{equation}
where  $ f\in L^1(\Omega) $ and $ \phi\in C^{0}(\mathbb{R}^{N})$,
he proved the solutions existence and some regularity results, under
the above assumptions.
 Aharouch and  Azroul \cite{LAEA}   studied  the problem \eqref{appp}
 in Oricz-sobolev spaces. They proved the existence of entropy solutions
 $ u \in W_0^{1,q}(\Omega)$. In the case of $ p = N$, they assume
in addition that there exists an N-function $H$ such that $H(t^{N})$ 
is equivalent to $M(t)$.
 Kbiri Alaoui,  Meskine and  Souissi \cite{KBMESO} proved the critical 
regularity $W_0^{1,\overline{q}}(\Omega)$ of solutions for nonlinear 
elliptic problems with right-hand side in $L\log^\alpha L(\Omega)$ and 
$\alpha\geq\frac{N-1}{N}$. Also they proved some regularity results 
when $\alpha<\frac{N-1}{N}$.

In this article, we consider the problem
\begin{equation}
\begin{gathered}
 - \operatorname{div}  a(x,u,\nabla u) + g(x,u,\nabla u) 
= f  - \operatorname{div}  \phi(u) \quad \text{in } \Omega, \\
 u  =   0\quad \text{on }  \partial\Omega,
\end{gathered} \label{eq1}
\end{equation}
where the right hand side is assumed to satisfy
\begin{equation}
f \in L^1(\Omega)\quad \text{and} \quad \phi \in C^{0}(\mathbb{R}^{N}).\label{ass4}
\end{equation}
We will study the strongly nonlinear boundary-value problem \eqref{eq1} 
 in the framework of variable exponent Sobolev spaces, we will prove 
the existence of entropy solutions and some $ \bar{q}(x)$-regularity results.

Recall that, since no growth hypothesis is assumed on  $\phi$,
the term $ \operatorname{div} \phi(v) $ may be meaningless, even as 
a distribution for a function $ v \in W^{1,r(x)}_0(\Omega)$, $r(x)>1 $
(see \cite{bocc1} and \cite{bocc2} for the case of constant exponent).

\begin{definition} \label{def}\rm
For  $ k>0 $ and $ s \in \mathbb{R} $, the truncation function $ T_k(.) $
is defined by
$$
T_k(s)=\begin{cases}
s &\text{if } |s|\leq k,\\
 k\frac{s}{|s|}&\text{if }|s|>k.
\end{cases}
$$
\end{definition}

 This article is organized as follows.
 In the section $2$ we recall some important definitions and results 
of variable exponent Lebesgue and Sobolev spaces. 
We introduce in the section $3$ some assumptions on $ a(x, s, \xi) $
and $ g(x, s, \xi) $ for which our problem has a solutions. 
The section $4$ contains some important lemmas useful to prove our
 main results. The section $5$ will be devoted to show the existence of 
entropy solutions for the problem  \eqref{eq1}, also we will give some 
important $ L^{\bar{q}(x)}$-regularity  results
for these solutions (the case $ p = 2 - 1/N $ and $ p = N $ are excluded).

\section{Preliminaries}

Let $ \Omega  $ be a bounded open subset of $\mathbb{R}^{N}$  ($N\geq 2$), 
we say that a real-valued continuous function $ p(.) $ is log-H\"{o}lder 
continuous in $ \Omega $ if
$$ 
|p(x) - p(y)| \leq \frac{C}{|\log|x-y||}  \quad\forall x,y\in \overline{\Omega} 
\text{ such that }  |x-y|<\frac{1}{2},
$$
with possible different constant $ C$. We denote
$$ 
C_{+}(\overline{\Omega})= \{\text{log-H\"{o}lder continuous function }
 p:\overline{\Omega}  \to \mathbb{R}  \text{ with } 
1 < p_{-} \leq p_{+} < N\},
$$
where
$$
p_{-} = \min\{p(x)  :  x\in \overline{\Omega}\}\quad
p_{+} = \max\{p(x)  :  x\in \overline{\Omega}\}.
$$

We define the variable exponent Lebesgue space for 
$ p\in C_{+}(\overline{\Omega}) $ by
$$ 
L^{p(x)}(\Omega) = \{u : \Omega \to \mathbb{R} \text{ measurable}:  
   \int_{\Omega} |u(x)|^{p(x)}  dx < \infty\},
$$
the space $L^{p(x)}(\Omega)$ under the norm
$$ 
\|u\|_{p(x)} = \inf  \big\{\lambda > 0 :
 \int_{\Omega} |\frac{u(x)}{\lambda}|^{p(x)}  dx \leq 1 \big\}
$$
is a uniformly convex Banach space, and therefore reflexive. 
We denote by $ L^{p'(x)}(\Omega) $ the conjugate space of $L^{p(x)}(\Omega) $ 
where $\frac{1}{p(x)}  + \frac{1}{p'(x)} = 1$ (see \cite{Fan2,zhao}).

\begin{proposition}[Generalized H\"{o}lder inequality \cite{Fan2,zhao}] 
\label{prop1}
(i) For any functions  $ u\in L^{p(x)}(\Omega) $ and $ v\in L^{p'(x)}(\Omega)$, we have
$$ 
|\int_{\Omega} u v dx | \leq \Big(\frac{1}{p_{-}} 
+ \frac{1}{p'_{-}}\Big) \|u\|_{p(x)}   \|v\|_{p'(x)} .
$$
(ii) For all  $p_1, p_{2}\in C_{+}(\overline{\Omega}) $ such that
$ p_1(x) \leq p_{2}(x) $ a.e. in $\Omega$, we have
 $ L^{p_{2}(x)}(\Omega) \hookrightarrow L^{p_1(x)}(\Omega) $
and the embedding is continuous.
\end{proposition}

\begin{proposition}[\cite{Fan2,zhao}] \label{prop2}
If we denote 
$$
 \rho(u) = \int_{\Omega} |u|^{p(x)}  dx \quad \forall  u\in L^{p(x)}(\Omega),
$$ 
then, the following assertions hold
\begin{itemize}
 \item[(i)] $\|u\|_{p(x)} < 1$ (resp, $= 1$, $> 1$) if and only if
 $\rho(u) < 1$  (resp,  $= 1$, $> 1$);
 \item[(ii)] $ \|u\|_{p(x)} >  1$ implies 
$ \|u\|_{p(x)}^{p_{-}} \leq  \rho(u) \leq \|u\|_{p(x)}^{p_{+}}  $, and 
 $ \|u\|_{p(x)} <  1 $ implies 
$\|u\|_{p(x)}^{p_{+}} \leq  \rho(u) \leq \|u\|_{p(x)}^{p_{-}}$;
 
\item[(iii)] $\|u\|_{p(x)} \to 0 $ if and only if $\rho(u) \to 0$, and 
$\|u\|_{p(x)} \to \infty$ if and only if $\rho(u) \to \infty$.
\end{itemize}
\end{proposition}

Now, we define the variable exponent Sobolev space by
$$ 
W^{1,p(x)}(\Omega) = \{ u\in L^{p(x)}(\Omega)   \text{ and }  
  |\nabla u|\in L^{p(x)}(\Omega) \},
$$
with the norm
$$ 
\|u\|_{1,p(x)}  = \|u\|_{p(x)} + \|\nabla u\|_{p(x)}  \quad 
\forall u \in W ^{1,p(x)}(\Omega).
$$
We denote by $ W_0^{1,p(x)}(\Omega)$ the closure of
$ C_0^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$,  and we define
the Sobolev exponent by $ p^{*}(x) = \frac{N p(x)}{N - p(x)}$ for 
$p(x) < N$.

\begin{proposition}[\cite{Fan2, Harjhast}] \label{prop3}
\begin{itemize}
\item[(i)] Assuming $ 1< p_{-}\leq p_{+} < \infty$, the spaces 
$ W^{1, p(x)}(\Omega) $ and $ W_0^{1, p(x)}(\Omega) $ are separable
and reflexive Banach spaces.

\item[(ii)]   If $q\in C_{+}(\bar{\Omega})$ and $q(x) < p^{*}(x) $ 
for any $x \in \Omega$, then the embedding 
$ W^{1, p(x)}_0(\Omega) \hookrightarrow\hookrightarrow L^{q(x)}(\Omega)$
is continuous and compact.

\item[(iii)]   Poincar\'{e} inequality:  there exists a constant $ C  > 0$, 
such that 
$$
 \| u\|_{p(x)}  \leq C  \|\nabla u\|_{p(x)} \quad \forall 
u\in W_0^{1,p(x)}(\Omega).
$$
\item[(vi)]  Sobolev-Poincar\'{e} inequality : 
there exists an other constant $ C > 0$, such that
$$ 
\| u\|_{p*(x)}  \leq C  \|\nabla u\|_{p(x)} \quad 
\forall u\in W_0^{1,p(x)}(\Omega).
$$
\end{itemize}
\end{proposition}

\begin{remark} \label{rmk2.1} \rm
By (iii) of Proposition \ref{prop3}, we deduce that $\|\nabla u\|_{p(x)}$ 
and $\|u\|_{1,p(x)}$ are equivalent norms in $W^{1,p(x)}_0(\Omega)$.
\end{remark}

\begin{definition}[\cite{DHHR}] \label{def2.1}\rm
We denote the dual of the Sobolev space $ W_0^{1,p(x)}(\Omega) $ by
$ W^{-1,p'(x)}(\Omega)$, and for each $ F\in W^{-1,p'(x)}(\Omega) $ 
there exists $ f_0, f_1, \ldots, f_{N} \in L^{p'(x)}(\Omega) $ such that
$ {F = f_0 + \sum_{i=1}^{N} \frac{\partial f_{i}}{\partial x_{i}} }$.
Moreover, for all $ u\in W_0^{1,p(x)}(\Omega) $ we have
$$ 
\langle F, u\rangle = \int_{\Omega} f_0u  dx
 - \sum_{i=1}^{N} \int_{\Omega} f_{i} \frac{\partial u}{\partial x_{i}} dx.  
$$
and we define a norm on the dual space by
$$
\|F\|_{-1,p'(x)} \simeq \sum_{i=0}^{N} \|f_{i}\|_{p'(x)}.
$$
\end{definition}

Now, we define
$$ 
T_0^{1,p(x)}(\Omega) := \{\text{measurable function $u$ such that } 
T_k(u)\in W_0^{1,p(x)}(\Omega)  \quad  \forall k > 0\}.
$$

\begin{proposition}\label{prop1b}
Let $ u \in T_0^{1,p(x)}(\Omega)$, there exists a unique measurable function 
$ v: \Omega\to \mathbb{R}^{N} $ such that
$$ 
v.\chi_{\{|u|\leq k\}}  = \nabla T_k(u) \quad \text{for a.e.  
 $x\in \Omega$  and for all } k> 0.
$$
We will define the gradient of $u$ as the function $v$, and we will 
denote it by $v=\nabla u$.
\end{proposition}

\begin{definition} \label{def2.2}\rm
A measurable function $ u $ is an entropy solution of the Dirichlet problem 
\eqref{eq1} if
\begin{gather*}
 T_k(u) \in W^{1,p(x)}_0(\Omega) \quad \forall   k>0,\\
\begin{aligned}
&\int_{\Omega}a(x,u,\nabla u) \nabla T_k(u-\varphi) dx 
 + \int_{\Omega}g(x,u,\nabla u) T_k(u-\varphi) dx \\
&\leq  \int_{\Omega} fT_k(u-\varphi) dx 
+ \int_{\Omega} \phi(u)\nabla T_k(u-\varphi) dx 
\end{aligned}
\end{gather*}
for all $\varphi \in W^{1,p(x)}_0(\Omega) \cap L^{\infty}(\Omega)$.
\end{definition}

\begin{lemma}\label{lem1.1}
Let $\lambda \in \mathbb{R}$ and let $u$ and $v$ be two functions which
are finite almost everywhere, and which belong to 
${\mathcal{T}}_0^{1,p(x)}(\Omega)$, then 
$$
\nabla(u+\lambda v)=\nabla u+ \lambda \nabla v\quad \text{ a.e. in } \Omega,
$$
where $\nabla u$, $\nabla v$ and $\nabla(u+\lambda v)$ are the gradients 
of $u$, $v$ and $u+\lambda v$ introduced in the Definition \ref{def2.2}.
\end{lemma}

\begin{proof}
Let $E_n=\{|u| \leq n\}\cap \{|v| \leq n\}$. We have
$T_n(u)=u$ and $T_n(v)=v$ in $E_n$, then for every $k>0$
$$
T_k(T_n(u)+\lambda T_n(v))=T_k(u+ \lambda v) \quad \text{a.e. in } E_n,
$$
and therefore, since both functions belong to $W_0^{1,p(x)}(\Omega)$,
\begin{equation}\label{2.13}
\nabla T_k(T_n(u)+\lambda T_n(v))=\nabla T_k(u+ \lambda v) \quad
 \text{a.e. in } E_n.
\end{equation}
Since $T_n(u)$ and $T_n(v)$ belong to $W_0^{1,p(x)}(\Omega)$, 
we have by using a classical property of the truncates functions in 
$W_0^{1,p(x)}(\Omega)$, and  the definition of $\nabla u$ and $\nabla v$,
\begin{align*}
    \nabla T_k(T_n(u)+\lambda T_n(v)) 
& = \chi_{\{|T_n(u)+\lambda T_n(v)|\leq k \}}(\nabla T_n(u)
 +\lambda \nabla T_n(v)) \\
& =  \chi_{\{|T_n(u)+\lambda T_n(v)|\leq k \}}(\nabla u.\chi_{\{|u|\leq n\}} 
 + \lambda \nabla v .\chi_{\{|v|\leq n\}}) 
\end{align*}
a.e. in  $\Omega$. Therefore,
\begin{equation}\label{2.14}
\nabla T_k(T_n(u)+\lambda T_n(v))
=\chi_{\{|u+\lambda v|\leq k\}} (\nabla u+ \lambda\nabla  v) \quad
 \text{a.e. in } E_n.
\end{equation}
On the other hand, by definition of $\nabla( u+ \lambda  v)$,
\begin{equation}\label{2.15}
\nabla T_k(u+\lambda v)=\chi_{\{|u+\lambda v|\leq k\}} \nabla( u+ \lambda  v)
 \quad \text{a.e. in } E_n.
\end{equation}
Putting together \eqref{2.13}, \eqref{2.14} and \eqref{2.15}, we obtain
\begin{equation}\label{2.16}
\chi_{\{|u+\lambda v|\leq k\}} \nabla( u+ \lambda  v)
=\chi_{\{|u+\lambda v|\leq k\}} (\nabla u+ \lambda\nabla  v) \quad 
\text{a.e. in } E_n.
\end{equation}
We have $\cup_{n \in \mathbb{N}} E_n$ 
(resp. $\cup_{k \in \mathbb{N}} \{ |u+\lambda v| \leq k \}$) 
differs at most from $\Omega$ by a set of zero Lebesgue measure, 
since $u$ and $v$ are almost everywhere finite, then \eqref{2.16}
 holds almost everywhere in $\Omega$. which conclude the proved of 
Lemma \ref{lem1.1}.

\section{Essential assumption}

 Let $ \Omega  $ be a bounded open subset of $\mathbb{R}^{N}$  ($N\geq 2$) 
 and $  p\in C_{+}(\bar{\Omega})$, we consider a Leray-Lions operator from 
$W_0^{1,p(x)}(\Omega)$ into its dual $W^{-1,p'(x)}(\Omega)$, defined by 
the formula
\begin{equation}
 Au = - \operatorname{div} \ a(x,u,\nabla u)
\end{equation}
where $  a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \to \mathbb{R}^{N}  $ 
is a Carath\'{e}odory function (measurable with respect to $x$ in $\Omega$ 
for every $(s, \xi)$ in $\mathbb{R} \times \mathbb{R}^{N}$, 
and continuous with respect to $(s, \xi)$ in $\mathbb{R} \times \mathbb{R}^{N}$ 
for almost every $x$ in $\Omega$) which satisfies the following conditions
\begin{gather}
|a(x,s,\xi)|\leq  \beta  (K(x) + |s|^{p(x) - 1} + |\xi|^{p(x) - 1}),\label{aq1}\\
a(x,s,\xi)\xi \geq \alpha |\xi|^{p(x)},\label{aq2}\\
[a(x,s,\xi) - a(x,s,\overline{\xi})](\xi - \overline{\xi}) > 0 
 \quad \text{for all $\xi \neq \overline{\xi}$  in $\mathbb{R}^{N}$},\label{aq3}
\end{gather}
for a.e. $x \in \Omega$, all $(s,\xi) \in \mathbb{R} \times \mathbb{R}^{N}$, 
where $ K(x) $ is a positive function lying in $ L^{p'(x)}(\Omega) $ and 
$ \alpha, \beta > 0$.

The nonlinear term $ g(x,s,\xi) $ is a Carath\'{e}odory function which satisfies
\begin{gather}
 g(x,s,\xi)s  \geq  0,\label{aq4} \\
|g(x,s,\xi)|\leq b(|s|)(c(x) + |\xi|^{p(x)}),\label{aq5}
\end{gather}
where $ b:\mathbb{R}^{+}\to \mathbb{R}^{+} $ is a continuous, 
nondecreasing function, and $ c: \Omega \to \mathbb{R}^{+} $ with 
$c\in L^1(\Omega)$.
We consider the problem
\begin{equation}
\begin{gathered}
 - \operatorname{div}  a(x,u,\nabla u) + g(x,u,\nabla u) 
= f - \operatorname{div}  \phi(u) \quad \text{in }  \Omega, \\
  u =    0\quad \text{on }  \partial\Omega,
\end{gathered} \label{aq7}
\end{equation}
with
\begin{equation}
f \in L^1(\Omega)\quad \text{and} \quad \phi \in C^{0}(\mathbb{R}^{N}).\label{aqq1}
\end{equation}
The symbol $\rightharpoonup$ will denote the weak convergence, 
and the constants  $ C_{i}$,   $i = 1, 2, \dots $ used in each steps 
of proof are independent.
\end{proof}

\section{Some technical Lemmas}

\begin{lemma}[\cite{EAM}] \label{lemp1} 
Let $ g\in L^{r(x)}(\Omega) $ and $ g_n\in L^{r(x)}(\Omega) $ with 
$ \|g_n\|_{r(x)} \leq C $ for $ 1< r(x)< \infty$. 
If $ g_n(x)\to g(x) $  a.e. on $ \Omega$, then $ g_n\rightharpoonup g $  
in $ L^{r(x)}(\Omega)$.
\end{lemma}

\begin{lemma}\label{lemp3}
Let $u\in W_0^{1,p(x)}(\Omega)$ then  $T_k(u)\in W_0^{1,p(x)}(\Omega)$ 
with $k>0$. Moreover, we have  
$T_k(u)\to u$  in $W_0^{1,p(x)}(\Omega)$   as $k\to \infty$.
\end{lemma}

\begin{proof} 
Let $k>0$ and $T_k:  \mathbb{R}  \to  \mathbb{R}$,
$$   
T_k(s)=  \begin{cases}
s & \text{if }   |s|\leq k,\\
 k.\text{sign}(s) & \text{if }  |s|>k,
\end{cases}
$$
then for all $ u\in W_0^{1,p(x)}(\Omega) $  we have 
$ T_k(u)\in W_0^{1, p(x)}(\Omega)$, and
\begin{align*}
& \int_{\Omega}|T_k(u)-u|^{p(x)}dx+\int_{\Omega}|\nabla T_k(u)-\nabla u|^{p(x)}dx\\
& =  \int_{\{|u|\leq k\}}|T_k(u)-u|^{p(x)}dx+\int_{\{|u|>k\}}|T_k(u)-u|^{p(x)}dx \\
& \quad+ \int_{\{|u|\leq k\}}|\nabla T_k(u)-\nabla u|^{p(x)}
 +\int_{\{|u|>k\}}|\nabla T_k(u)-\nabla u|^{p(x)}dx\\
&  =  \int_{\{|u|>k\}}|T_k(u)-u|^{p(x)}dx +\int_{\{|u|>k\}}|\nabla u|^{p(x)}dx  .
\end{align*}
Since $T_k(u)\to u$ as $k\to\infty$ and by using the dominated convergence 
theorem, we have
$$ 
\int_{\{|u|>k\}}|T_k(u)-u|^{p(x)}dx +\int_{\{|u|>k\}}|\nabla u|^{p(x)}dx\to 0
\quad \text{ as } k\to \infty.
$$
Finally $\|T_k(u)-u\|_{W_0^{1,p(x)}(\Omega)}\to 0$ as $k\to \infty$.
\end{proof}

\begin{lemma}[\cite{BenWit}] \label{lemp5}
Let $ p(\cdot) $ be a continuous function in $ C_{+}(\overline{\Omega}) $ and 
$ u $ a function in $ W_0^{1,p(x)}(\Omega)$. 
Suppose $ {2- \frac{1}{N} < p_{-} \leq p_{+} < N} $, and that there exists 
a constant $ c_1 $ such that
$$ 
\int_{\{k \leq |u| \leq k+1\}} |\nabla u|^{p(x)} dx \leq c_1\quad \forall k > 0.
$$
Then there exists a constant $ c_{2} > 0$, depending on $ c_1$, such that
$$ 
\|u\|_{1,q(x)} \leq c_{2},
$$
for all continuous functions $ q(\cdot) $ on $ \overline{\Omega} $ satisfying
$$ 
1 \leq q(x) < \frac{N(p(x) - 1)}{N - 1}\quad \text{for all }
 x \in \overline{\Omega}.
$$
\end{lemma}

\begin{lemma}\label{lemp4}
Assume  \eqref{aq1}-\eqref{aq3}, and let $ (u_n)_n $ 
be a sequence in $ W_0^{1,p(x)}(\Omega) $ such that $ u_n \rightharpoonup u $ 
in $ W_0^{1,p(x)}(\Omega) $ and
\begin{equation}
\int_{\Omega} [a(x, u_n, \nabla u_n) - a(x, u_n, \nabla u)]\nabla (u_n  -  u ) dx
 \to 0,\label{eqq1}
\end{equation}
then $ u_n\to u \quad \text{in } \quad W_0^{1,p(x)}(\Omega)$ for a subsequence.
\end{lemma}

\begin{proof}
Let $ D_n = [a(x,u_n,\nabla u_n) - a(x,u_n,\nabla u)]\nabla(u_n - u)$, 
thanks to \eqref{aq3} we have $ D_n $ is a positive function, and by 
\eqref{eqq1}, $ D_n\to 0  $ in $ L^1(\Omega) $   as    $n \to\infty$.

Since $ u_n \rightharpoonup u $  in $ W_0^{1,p(x)}(\Omega) $ then 
$ u_n \to\ u $ a.e. in $ \Omega$, and since $ D_n\to 0 $ a.e. in
$ \Omega$, there exists a subset $ B $ in $ \Omega $ with measure zero such 
that  for all $x\in \Omega\backslash B$,
$$ 
|u(x)|<\infty,\quad |\nabla u(x)|<\infty,\quad K(x) < \infty,\quad  
u_n \to\ u,\quad  D_n\to 0.
$$
Taking $ \xi_n = \nabla u_n $ and $  \xi = \nabla u$, we have
\begin{align*}
D_n(x) 
& = [a(x,u_n,\xi_n) - a(x,u_n,\xi)](\xi_n - \xi) \\
& = a(x,u_n,\xi_n)\xi_n + a(x,u_n,\xi)\xi - a(x,u_n,\xi_n)\xi 
 - a(x,u_n,\xi)\xi_n \\
& \geq \alpha |\xi_n|^{p(x)} + \alpha |\xi|^{p(x)} 
 - \beta (K(x)+ |u_n|^{p(x) -1} + |\xi_n|^{p(x)-1})|\xi| \\
&\quad - \beta (K(x)+ |u_n|^{p(x)-1} + |\xi|^{p(x)-1})|\xi_n| \\
& \geq  \alpha |\xi_n|^{p(x)} - C_{x}(1+ |\xi_n|^{p(x)-1} + |\xi_n|),
\end{align*}
where $ C_{x} $ depending on $ x$, without dependence on $ n$. 
(since $ u_n(x)\to u(x) $ then $ (u_n)_n $ is bounded), we obtain
$$ 
D_n(x) \geq |\xi_n|^{p(x)} \big(\alpha - \frac{C_{x}}{|\xi_n|^{p(x)}} 
-\frac{C_{x}}{|\xi_n|} - \frac{C_{x}}{|\xi_n|^{p(x)-1}}\big),
$$
by the standard argument $ (\xi_n)_n $ is bounded almost everywhere in
 $ \Omega$, (Indeed, if $ |\xi_n|\to \infty $ in a measurable subset 
$ E\in \Omega $  then
$$ 
\lim_{n \to \infty}\int_{\Omega} D_n(x) dx 
\geq \lim_{n \to \infty}\int_{E} |\xi_n|^{p(x)} 
\big(\alpha - \frac{C_{x}}{|\xi_n|^{p(x)}} -\frac{C_{x}}{|\xi_n|} 
- \frac{C_{x}}{|\xi_n|^{p(x)-1}}\big) dx =  \infty, 
$$
 which is absurd since $ D_n\to 0  $ in $ L^1(\Omega) $).

Let $ \xi^{*} $ an accumulation point of $ (\xi_n)_n$, we have 
$ |\xi^{*}|<\infty $ and by the continuity of $ a(.,.,.) $ we obtain,
$$
[a(x,u(x),\xi^{*}) - a(x,u(x),\xi)](\xi^{*} - \xi) = 0,
$$
thanks to \eqref{aq3} we have $\xi^{*} = \xi$, the uniqueness of the 
accumulation point implies that $\nabla u_n\to \nabla u$ a.e. in $\Omega$.
since $ (a(x,u_n,\nabla u_n))_n $ is bounded in $ (L^{p'(x)}(\Omega))^{N} $  
and  $ a(x,u_n,\nabla u_n) \to a(x,u,\nabla u) $ a.e. in $\Omega$, by the 
Lemma $\ref{lemp1}$, we can establish that
$$ 
a(x,u_n,\nabla u_n) \rightharpoonup a(x,u,\nabla u)\quad \text{in }
 (L^{p'(x)}(\Omega))^{N}.
$$
Let us taking $ \bar{y}_n = a(x,u_n,\nabla u_n)\nabla u_n$   and   
$\bar{y} = a(x,u,\nabla u)\nabla u$, then $ \bar{y}_n \to \bar{y} $ in 
$ L^1(\Omega)$, according to the condition $ \eqref{aq2} $ we have
$$
\alpha|\nabla u_n|^{p(x)} \leq  a(x,u_n,\nabla u_n)\nabla u_n,
$$
Let $ z_n = \nabla u_n , z = \nabla u $ and 
$  y_n = \frac{\bar{y}_n}{\alpha}$,   
$y = \frac{\bar{y}}{\alpha}$, in view of the Fatou Lemma, we obtain
$$
\int_{\Omega} 2.y  dx \leq \liminf_{n\to \infty} 
\int_{\Omega} (y_n + y - |z_n - z|^{p(x)})dx,
$$
then $ 0 \leq -\limsup_{n\to \infty}\int_{\Omega}|z_n - z|^{p(x)} dx $, and since
$$
0 \leq \liminf_{n\to \infty}\int_{\Omega}|z_n - z|^{p(x)} dx 
\leq \limsup_{n\to \infty}\int_{\Omega}|z_n - z|^{p(x)} dx \leq 0,
$$
 it follows that $ { \int_{\Omega} |\nabla u_n -\nabla u|^{p(x)} dx \to 0} $ 
as $ n\to \infty$, and we get
$$ 
\nabla u_n \to\nabla u \quad in \quad (L^{p(x)}(\Omega))^{N}
$$
we deduce that
$$ 
u_n \to u \quad \text{in } W_0^{1,p(x)}(\Omega),
$$
which completes our proof.
\end{proof}

Now, we consider $ \phi_n(s) = \phi(T_n(s)) $ with 
$ \phi \in C^{0}(\mathbb{R}^{N}) $ and 
\[
 {g_n(x,s,\xi) = \frac{g(x,s,\xi)}{1 + \frac{1}{n}|g(x,s,\xi)|}} 
\]
 such that 
$ g(x,s,\xi) $ satisfies $ \eqref{aq4}-\eqref{aq5}$, note that
$$
 g_n(x,s,\xi) s\geq 0, \quad  |g_n(x,s,\xi)|\leq |g(x,s,\xi)|,\quad 
|g_n(x,s,\xi)|\leq n \quad \forall n\in \mathbb{N}^{*}.
$$
We define the operator $ G_n : W_0^{1,p(x)}(\Omega) \to W^{-1,,p'(x)}(\Omega)$, 
by
$$
\langle G_n u,v \rangle = \int_{\Omega} g_n(x,u,\nabla u)v  dx  \quad 
 \forall v\in W_0^{1,p(x)}(\Omega).
$$
Thanks to the H\"{o}lder inequality, we have that for all 
$  u, v \in W_0^{1,p(x)}(\Omega)$,
\begin{equation}
\begin{aligned}
&\big|\int_{\Omega} g_n(x,u,\nabla u)v  dx\big|\\
& {\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  
\|g_n(x,u,\nabla u)\|_{p'(x)}\|v\|_{p(x)}} \\
&{ \leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big) 
 \Big(\int_{\Omega} |g_n(x,u,\nabla u)|^{p'(x)} dx + 1\Big)^{\frac{1}{p'_{-}}}
 \|v\|_{1,p(x)}} \\
& {\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  
\Big(\int_{\Omega} n^{p'(x)} dx + 1\Big)^{\frac{1}{p'_{-}}}\|v\|_{1,p(x)}} \\
& {\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  \big( n^{p'_{+}}.\operatorname{meas}(\Omega) + 1\big)^{\frac{1}{p'_{-}}}\|v\|_{1,p(x)}} \\
& \leq C_0\|v\|_{1,p(x)} ,
\end{aligned}\label{aq10}
\end{equation}
and we define the operator 
$ R_n = \operatorname{div}  \phi_n :  W_0^{1,p(x)}(\Omega) 
\to W^{-1,p'(x)}(\Omega)$, such that
$$ 
\langle R_n(u),v \rangle = \langle \operatorname{div}  \phi_n(u),v \rangle 
= - \int_{\Omega} \phi_n(u)\nabla v  dx  \quad 
\forall u,v \in W_0^{1,p(x)}(\Omega),
$$
we have
\begin{equation}
\begin{aligned}
{\big|\int_{\Omega} \phi_n(u)\nabla v  dx \big|} 
&{\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  
 \|\phi_n(u)\|_{p'(x)}\|\nabla v\|_{p(x)}} \\
& {\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  
 \Big(\int_{\Omega} |\phi_n(u)|^{p'(x)} dx + 1\Big)^{\frac{1}{p'_{-}}}
 \|v\|_{1,p(x)}} \\
& {\leq \big(\frac{1}{p_{-}}  +  \frac{1}{p'_{-}}\big)  
 \big(\sup_{|s|\leq n} (|\phi(s)| + 1)^{p'_{+}}
\operatorname{meas}(\Omega) + 1 \big)^{1/p'_{-}}\|v\|_{1,p(x)}} \\
& \leq C_1 \|v\|_{1,p(x)}.
\end{aligned}\label{aq11}
\end{equation}

\begin{lemma}\label{lem1}
The operator $ B_n = A  +  G_n +  R_n $ is  pseudo-monotone from 
$ W_0^{1,p(x)}(\Omega) $ into $ W^{-1,p'(x)}(\Omega)$. Moreover, 
$B_n$ is coercive in the following sense
$$
\frac{\langle B_n v,v\rangle}{\|v\|_{1,p(x)}} \to 
+ \infty \quad \text{as} \quad \|v\|_{1,p(x)} \to 
+ \infty \quad \text{for}\quad v \in W_0^{1,p(x)}(\Omega).
$$
\end{lemma}

\begin{proof}
Using H\"{o}lder's inequality and the growth condition  \eqref{aq1},
we can show that the operator $ A $ is bounded, and by using \eqref{aq10}
 and \eqref{aq11} we conclude that $ B_n $ bounded. 
For the coercivity, we have for any $ u\in W_0^{1,p(x)}(\Omega)$,
\begin{align*}
{\langle B_n u,u\rangle} 
& = {\langle A u,u\rangle  + \langle G_n u,u\rangle + \langle R_n u,u\rangle} \\
& = {\int_{\Omega} a(x,u,\nabla u)\nabla u dx 
  + \int_{\Omega} g_n(x,u,\nabla u)u dx 
  - \int_{\Omega} \phi_n(u)\nabla u dx} \\
& \geq {\alpha\int_{\Omega} |\nabla u|^{p(x)} dx 
  - \big(\frac{1}{p_{-}} + \frac{1}{p'_{-}}\big)\|\phi_n(u)\|_{p'(x)} \|\nabla u\|_{p(x)} } \\
& \geq {\alpha\|\nabla u\|_{p(x)}^{\delta} - C_1.\|u\|_{1,p(x)}}
  \quad (\text{using} \eqref{aq11} ) \\
& \geq {\alpha'\|u\|_{1,p(x)}^{\delta} - C_1.\|u\|_{1,p(x)} ,}
  \quad (\text{using the Poincar\'{e} inequality})
\end{align*}
with
$$ 
\delta = \begin{cases}
p_{-} & \text{if }  \|\nabla u\|_{p(x)} > 1, \\
p_{+} & \text{if }  \|\nabla u\|_{p(x)} \leq 1,
 \end{cases}
$$
then, we obtain
$$
\frac{\langle B_n u,u\rangle}{\|u\|_{1,p(x)}} \to + \infty \quad 
\text{as } \|u\|_{1,p(x)} \to + \infty.
$$
It remains to show that $B_n$ is pseudo-monotone. 
Let $ (u_k)_k $ a sequence in $ W_0^{1,p(x)}(\Omega) $ such that
\begin{equation}
\begin{gathered}
 u_k \rightharpoonup u \quad \text{in }     W_0^{1,p(x)}(\Omega), \\
  B_nu_k \rightharpoonup \chi \quad \text{in }     W^{-1,p'(x)}(\Omega), \\
 \limsup_{k\to \infty}  \langle B_nu_k,u_k\rangle \leq \langle \chi,u\rangle.
\end{gathered}\label{aq12}
\end{equation}
We will prove that
$$
\chi = B_nu \quad  \text{and} \quad 
\langle B_nu_k,u_k\rangle \to \langle \chi,u\rangle \quad  \text{as } 
 k \to + \infty.
$$
Firstly, since 
$W_0^{1,p(x)}(\Omega)\hookrightarrow\hookrightarrow L^{p(x)}(\Omega)$, 
then $ u_k\to u \text{ in } L^{p(x)}(\Omega)  $ for a subsequence 
still denoted $ (u_k)_k $.

We have $ (u_k)_k $ is a bounded sequence in $ W_0^{1,p(x)}(\Omega)$, 
then by the growth condition $ (a(x,u_k,\nabla u_k))_k $ is bounded 
in $ (L^{p'(x)}(\Omega))^{N}$, therefore, there exists a function 
$\varphi\in (L^{p'(x)}(\Omega))^{N}$ such that
\begin{equation}
a(x,u_k,\nabla u_k) \rightharpoonup \varphi \quad \text{in } 
 (L^{p'(x)}(\Omega))^{N} \text{ as }  k\to \infty.\label{aq13}
\end{equation}
Similarly, since $ (g_n(x,u_k,\nabla u_k))_k $ is bounded in
 $ L^{p'(x)}(\Omega)$, then there exists a function 
$ \psi_n\in L^{p'(x)}(\Omega) $ such that
\begin{equation}
g_n(x,u_k,\nabla u_k) \rightharpoonup \psi_n \quad \text{in }
 L^{p'(x)}(\Omega)  \text{ as }  k\to \infty,\label{aq14}
\end{equation}
and since $ \phi_n = \phi\circ T_n $ is a bounded continuous function 
and $ u_k\to u$  in $L^{p(x)}(\Omega)$, it follows
\begin{equation}
\phi_n(u_k) \to \phi_n(u) \quad \text{in }  (L^{p'(x)}(\Omega))^{N} 
  \text{ as }  k \to \infty. \label{aq15}
\end{equation}
For all $v\in W_0^{1,p(x)}(\Omega)$, we have
\begin{equation}
\begin{aligned}
\langle\chi,v\rangle 
& = \lim_{k\to \infty} \langle B_nu_k ,v\rangle\\
& = \lim_{k\to \infty} \int_{\Omega} a(x,u_k,\nabla u_k)\nabla v  dx  
 + \lim_{k\to \infty}\int_{\Omega} g_n(x,u_k,\nabla u_k)v  dx \\
&\quad - \lim_{k\to \infty} \int_{\Omega} \phi_n(u_k) \nabla v  dx\\
& = {\int_{\Omega} \varphi\nabla v  dx + \int_{\Omega} \psi_n v  dx 
 - \int_{\Omega} \phi_n(u) \nabla v  dx}.
\end{aligned}\label{aq16}
\end{equation}
Using  \eqref{aq12} and \eqref{aq16}, we obtain
\begin{equation}
\begin{aligned}
&\limsup_{k\to \infty}  \langle B_n(u_k),u_k\rangle \\
& = \limsup_{k \to\infty} \Big\{\int_{\Omega} a(x,u_k,\nabla u_k)\nabla u_k  dx 
 + \int_{\Omega} g_n(x,u_k,\nabla u_k) u_k dx 
 - \int_{\Omega} \phi_n(u_k)\nabla u_k  dx \Big\}\\
&\leq \int_{\Omega} \varphi\nabla u  dx + \int_{\Omega} \psi_n u dx 
- \int_{\Omega} \phi_n(u) \nabla u  dx,
\end{aligned}\label{aq18}
\end{equation}
thanks to  \eqref{aq14} and \eqref{aq15}, we have
\begin{equation}
\int_{\Omega} g_n(x,u_k,\nabla u_k) u_k dx \to \int_{\Omega} \psi_n u dx,
\quad  
\int_{\Omega} \phi_n(u_k) \nabla u_k dx \to \int_{\Omega} \phi_n(u) \nabla u dx;
\label{aq17}
\end{equation}
therefore,
\begin{equation}
\limsup_{k\to \infty} \int_{\Omega} a(x,u_k,\nabla u_k)\nabla u_k  dx 
\leq \int_{\Omega} \varphi\nabla u  dx.\label{aq19}
\end{equation}
On the other hand, using \eqref{aq3}, we have
\begin{equation}
\int_{\Omega} (a(x,u_k,\nabla u_k) 
- a(x,u_k,\nabla u))(\nabla u_k - \nabla u) dx \,.
\geq 0,\label{aq21}
\end{equation}
Then
\begin{align*}
&\int_{\Omega} a(x,u_k,\nabla u_k)\nabla u_k   dx \\
&\geq - \int_{\Omega} a(x,u_k,\nabla u)\nabla u   dx 
+\int_{\Omega} a(x,u_k,\nabla u_k)\nabla u  dx  
+  \int_{\Omega} a(x,u_k,\nabla u)\nabla u_k   dx,
\end{align*}
and by \eqref{aq13}, we get
$$ 
\liminf_{k\to \infty} \int_{\Omega} a(x,u_k,\nabla u_k)\nabla u_k   dx 
\geq  \int_{\Omega} \varphi \nabla u  dx,
$$
this implies, thanks to \eqref{aq19}, that
\begin{equation}
\lim_{k\to \infty} \int_{\Omega} a(x,u_k,\nabla u_k)\nabla u_k   dx 
= \int_{\Omega} \varphi \nabla u  dx.\label{aq22}
\end{equation}
By combining of \eqref{aq16}, \eqref{aq17} and \eqref{aq22}, we deduce that
$$ 
\langle B_nu_k , u_k\rangle  \to \langle \chi , u\rangle \quad  \text{as } 
  k \to +\infty.
$$
Now, by \eqref{aq22}  we can obtain
$$ 
\lim_{k \to +\infty} \int_{\Omega} 
(a(x,u_k,\nabla u_k) - a(x,u_k,\nabla u))(\nabla u_k - \nabla u) dx = 0,
$$
in view of the Lemma \ref{lemp4}, we obtain
$$  
u_k  \to u , \quad W_0^{1,p(x)}(\Omega), \quad  \nabla u_k  \to \nabla u\quad 
 \text{a.e. in }    \Omega,
$$
then
$$ 
a(x,u_k, \nabla u_k) \rightharpoonup a(x,u, \nabla u),\quad 
\phi_n(u_k) \to  \phi_n(u) \quad \text{in } (L^{p'(x)}(\Omega))^{N},
$$
and
$$ 
g_n(x,u_k, \nabla u_k) \rightharpoonup g_n(x,u, \nabla u) \quad \text{in } 
 L^{p'(x)}(\Omega),
$$
we deduce that $ \chi = B_n u$, which completes the proof.
\end{proof}

\section{Main results}

 In the sequel we assume that $\Omega$ is an open bounded subset of
 $\mathbb{R}^{N}$ $(N\geq 2)$, and let $ p(.)\in C_{+}(\overline{\Omega})$. 
We will prove the following existence results

\begin{theorem}\label{thm1}
 Assuming that \eqref{aq1}-\eqref{aq5} hold, 
$ p(.)\in C_{+}(\overline{\Omega})$,  $f\in L^1(\Omega) $ and 
$ \phi \in C^{0}(\mathbb{R}^{N})$, then the problem
\begin{equation}
\begin{gathered}
T_k(u) \in W_0^{1,p(x)}(\Omega) \quad \forall k > 0,\\
\begin{aligned}
&\int_{\Omega} a(x,u,\nabla u)\nabla T_k(u-\varphi)  dx  
 + \int_{\Omega} g(x,u,\nabla u) T_k(u-\varphi)  dx \\
& \leq \int_{\Omega} fT_k(u-\varphi)  dx 
 + \int_{\Omega} \phi(u)\nabla T_k(u-\varphi)  dx, \quad
 \forall \varphi \in W_0^{1,p(x)}(\Omega) \cap L^{\infty}(\Omega),
\end{aligned}
\end{gathered}\label{aq8}
\end{equation}
has at least one solution.
\end{theorem}

The above theorem is prove in the following 5 steps.

\subsection*{Step 1: Approximate problems}
 Let $ (f_n)_n $ be a sequence in $ W^{-1,p'(x)}(\Omega)\cap L^1(\Omega) $ 
such that $ f_n \to f $ in $ L^1(\Omega) $ with $ \|f_n\|_1 \leq \|f\|_1 $
and we consider the approximate problem
\begin{equation}
\begin{gathered}
Au_n + g_n(x,u_n,\nabla u_n) = f_n - \operatorname{div} \phi_n(u_n)\\
u_n\in W_0^{1,p(x)}(\Omega), 
\end{gathered}\label{aq9}
\end{equation}
with $ \phi_n(s) = \phi(T_n(s)) $ and 
$ {g_n(x,s,\xi) = \frac{g(x,s,\xi)}{1 + \frac{1}{n}|g(x,s,\xi)|}}$. 
In view of the Lemma \ref{lem1}, there exists at least one weak 
solution $ u_n \in W_0^{1,p(x)}(\Omega) $ of the problem  \eqref{aq9}, 
(cf.  \cite{lions}).

\subsection*{Step 2: A priori estimates}
Taking $ T_k(u_n) $ as a test function in \eqref{aq9}, we obtain
\begin{equation}
\begin{aligned}
&\int_{\Omega} a(x,u_n, \nabla u_n)\nabla T_k(u_n)  dx 
+ \int_{\Omega} g_n(x,u_n, \nabla u_n) T_k(u_n)  dx \\
&= \int_{\Omega} f_n T_k(u_n)  dx + \int_{\Omega} \phi_n(u_n)\nabla T_k(u_n)  dx.
\end{aligned}\label{aq23}
\end{equation}
Thanks to \eqref{aq2} and Young's inequality, we obtain
\begin{equation}
\begin{aligned}
&{\alpha\int_{\Omega} |\nabla T_k(u_n)|^{p(x)} dx} \\
& {\leq \int_{\Omega} a(x,T_k(u_n), \nabla T_k(u_n))\nabla T_k(u_n)  dx 
 + \int_{\Omega} g_n(x,u_n, \nabla u_n) T_k(u_n)  dx} \\
& {= \int_{\Omega} f_n T_k(u_n)  dx 
 + \int_{\Omega} \phi_n(T_k(u_n))\nabla T_k(u_n)  dx} \\
& {\leq k\int_{\Omega} |f_n| dx 
 + \int_{\Omega} \frac{|\phi_n(T_k(u_n))|}{(\frac{\alpha}{2} p(x))^{\frac{1}{p(x)}}}(\frac{\alpha}{2} p(x))^{\frac{1}{p(x)}}|\nabla T_k(u_n)|  dx} \\
& {\leq k\|f_n\|_1 + \int_{\Omega} \frac{|\phi_n(T_k(u_n))|^{p'(x)}}{p'(x)
 (\frac{\alpha}{2} p(x))^{\frac{p'(x)}{p(x)}}}  dx 
 +\int_{\Omega}\frac{\frac{\alpha}{2} p(x)|\nabla T_k(u_n)|^{p(x)}}{p(x)}  dx} \\
& {\leq k\|f\|_1 + C_{2}\int_{\Omega} |\phi_n(T_k(u_n))|^{p'(x)}  dx 
 + \frac{\alpha}{2}\int_{\Omega}|\nabla T_k(u_n)|^{p(x)}  dx,}
\end{aligned}\label{aq24}
\end{equation}
and since
\begin{align*}
{\int_{\Omega} |\phi_n(T_k(u_n))|^{p'(x)}  dx} 
& {\leq \int_{\Omega}  \sup_{|s|\leq k}|\phi_n(s)|^{p'(x)}  dx} \\
& {\leq \int_{\Omega}   \sup_{|s|\leq n}|\phi(s)|^{p'(x)}  dx} \\
& {\leq \big(\sup_{|s|\leq n}|\phi(s)| +1\big)^{p'_{+}}.\operatorname{meas}(\Omega),}
\end{align*}
by \eqref{aq24}, we obtain
$$ 
\frac{\alpha}{2}\|\nabla T_k(u_n)\|_{p(x)}^{\gamma}
\leq \frac{\alpha}{2}\int_{\Omega} |\nabla T_k(u_n)|^{p(x)} dx 
\leq k\|f\|_1 + C_{3},
$$
with
$$ 
\gamma = \begin{cases}
p_{+} & \text{if }   \|\nabla T_k(u_n)\|_{p(x)} \leq 1, \\
p_{-} & \text{if }   \|\nabla T_k(u_n)\|_{p(x)} > 1, 
\end{cases}
$$
we deduce that
\begin{equation}
\|\nabla T_k(u_n)\|_{p(x)} \leq C_{4}k^{\frac{1}{\gamma}}\quad 
\text{for all }   k\geq 1,\label{aq25}
\end{equation}
where $C_{4}$ is a  constant that does not depend on $k$.

Now, we  show that $(u_n)_n$ is a Cauchy sequence in measure. 
Indeed, we have
\begin{align*}
k \operatorname{meas}\{|u_n|> k\} 
& =  \int_{\{|u_n|> k\}} |T_k(u_n)|  dx \leq \int_{\Omega}  |T_k(u_n)|  dx \\
& \leq { \big(\frac{1}{p_{-}} 
 + \frac{1}{p'_{-}}\big)\|1\|_{p'(x)}\|T_k(u_n)\|_{p(x)}} \\
& \leq  {\big(\frac{1}{p_{-}} + \frac{1}{p'_{-}}\big) 
 (\operatorname{meas}(\Omega)+1)^{\frac{1}{p'_{-}}}\|T_k(u_n)\|_{p(x)} }\\
& \leq  C_{5} k^{\frac{1}{\gamma}}, 
\end{align*} 
according to the Poincar\'{e} inequality  and \eqref{aq25}. 
Therefore,
\begin{equation}
\operatorname{meas}\{|u_n|> k\} \leq C_{5}\frac{1}{k^{1-\frac{1}{\gamma}}} \to 0 
\quad \text{as } k \to \infty.\label{aq26}
\end{equation}
Since for all $\delta > 0 $,
\begin{align*}
&\operatorname{meas} \{|u_n - u_{m}|>\delta\} \\
&\leq \operatorname{meas} \{|u_n|>k\} + \operatorname{meas} \{|u_{m}|>k\} 
+\operatorname{meas} \{|T_k(u_n) - T_k(u_{m})|>\delta\},
\end{align*}
 using \eqref{aq26}, we get that for all $ \varepsilon > 0$, there exists
 $ k_0> 0 $ such that
\begin{equation}
\operatorname{meas}\{|u_n|> k\} \leq \frac{\varepsilon}{3}, \quad 
\operatorname{meas}\{|u_{m}|> k\} \leq \frac{\varepsilon}{3}\quad 
 \forall k \geq k_0(\varepsilon), \label{aq27}
\end{equation}
On the other hand, by \eqref{aq25}, the sequence $(T_k(u_n))_n$ 
is bounded in $W_0^{1,p(x)}(\Omega)$, then there exists a subsequence 
still denoted $(T_k(u_n))_n$ such that
$$ 
T_k(u_n) \rightharpoonup  \eta_k \quad \text{in } W_0^{1,p(x)}(\Omega)   
 \quad \text{as }  n\to \infty.
$$
and by the compact embedding, we obtain
$$ 
T_k(u_n) \to  \eta_k \quad \text{in }  L^{p(x)}(\Omega) \text{  and a.e.   in  }
\Omega.
$$
Therefore, we can assume that  $(T_k(u_n))_n$ is a Cauchy sequence in
 measure in $\Omega$, then for all $ k> 0 $ and $ \delta, \varepsilon > 0 $
 there exists  $n_0 = n_0(k, \delta,\varepsilon) $  such that
\begin{equation}
\operatorname{meas} \{|T_k(u_n)-T_k(u_{m})|> \delta\} 
\leq \frac{\varepsilon}{3} \quad  \forall  m,n \geq n_0. \label{aq281}
\end{equation}
Combining \eqref{aq27} and \eqref{aq281}, we obtain
that for all $\delta, \varepsilon > 0$, there exists 
$n_0 = n_0(\delta,\varepsilon)$ such that 
$$
\operatorname{meas}\{|u_n-u_{m}|> \delta\} \leq \varepsilon\quad 
\forall n, m \geq n_0,
$$
it follows that $(u_n)_n$ is a Cauchy sequence in measure, then there exists 
a subsequence still denoted $ (u_n)_n $ such that
$$ 
u_n \to u \quad \text{a.e. in } \Omega.
$$
We obtain
\begin{equation}
\begin{gathered}
T_k(u_n) \rightharpoonup T_k(u) \quad \text{in }   W_0^{1,p(x)}(\Omega) \\
T_k(u_n) \to T_k(u) \quad \text{in $L^{p(x)}(\Omega)$ and  a.e. in } \Omega.
\end{gathered}\label{aq28}
\end{equation}

\subsection*{Step 3: Convergence of the gradient}
In the sequel, we denote by $ \varepsilon_{i}(n)$  
$i = 1, 2, \ldots  $  various functions of real numbers which converge 
to $ 0 $ as $ n $ tends to infinity.
Let $ \varphi_k(s) = s\exp(\gamma s^{2}) $ where 
$ \gamma = \big(\frac{b(k)}{2\alpha}\big)^{2}$, it is obvious that
$$ 
\varphi'_k(s) - \frac{b(k)}{\alpha}|\varphi_k(s)| 
\geq \frac{1}{2}\quad \quad \forall s\in \mathbb{R},
$$
we consider $ h> k > 0 $ and $ M = 4k + h$, we set
$$
 \omega_n = T_{2k}(u_n - T_{h}(u_n) + T_k(u_n) - T_k(u)).
$$
Taking $ \varphi_k(\omega_n) $ as a test function in \eqref{aq9}, 
we obtain
\begin{align*} 
&\int_{\Omega} a(x,u_n, \nabla u_n) \varphi'_k(\omega_n)\nabla \omega_n   dx  
 +  \int_{\Omega} g_n(x,u_n, \nabla u_n) \varphi_k(\omega_n)   dx\\
& = \int_{\Omega} f_n\varphi_k(\omega_n)   dx  
+   \int_{\Omega} \phi_n(u_n)\varphi'_k(\omega_n)\nabla \omega_n   dx,
\end{align*}
it is easy to see that $ \nabla \omega_n = 0  $ on $ \{|u_n|> M\}$, 
and since $ g_n(x,u_n, \nabla u_n) \varphi_k(\omega_n) \geq 0  $ on 
$ \{|u_n| > k\}$, we have
 \begin{equation}
\begin{aligned}
&{\int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n))\varphi'_k(\omega_n)
 \nabla \omega_n   dx + \int_{\{|u_n| \leq k\}} g_n(x,u_n, \nabla u_n)
 \varphi_k(\omega_n)   dx}  \\\
& \leq \int_{\Omega} f_n\varphi_k(\omega_n)   dx 
+ \int_{\{|u_n| \leq M\}} \phi_n(T_{M}(u_n))\varphi'_k(\omega_n)\nabla \omega_n  
 dx.
\end{aligned}\label{aq29}
\end{equation}
We have
\begin{equation}
\begin{aligned}
&\int_{\Omega}  a(x,T_{M}(u_n), \nabla T_{M}(u_n))\varphi'_k(\omega_n)
 \nabla \omega_n   dx  \\
& =  \int_{\{|u_n|\leq k\}} a(x,T_k(u_n), \nabla T_k(u_n))\varphi'_k(\omega_n)
 \nabla T_{2k}(u_n - T_k(u)) dx \\
&\quad   +  \int_{\{|u_n| > k\}} a(x,T_{M}(u_n), \nabla T_{M}(u_n))
 \varphi'_k(\omega_n)\nabla T_{2k}(u_n-T_{h}(u_n)\\
&\quad +T_k(u_n)  - T_k(u)) dx.
\end{aligned}\label{aq30}
\end{equation}
On the one hand, since $ |u_n - T_k(u)|\leq 2k $ on $ \{|u_n|\leq k\}$, we have
\begin{equation}
\begin{aligned}
&{\int_{\{|u_n|\leq k\}} a(x,T_k(u_n), \nabla T_k(u_n))\varphi'_k(\omega_n)
 \nabla T_{2k}(u_n - T_k(u)) dx}  \\
& = \int_{\{|u_n|\leq k\}} a(x,T_k(u_n), \nabla T_k(u_n))
 \varphi'_k(\omega_n )\nabla (T_k(u_n) - T_k(u)) dx \\
& = \int_{\Omega} a(x,T_k(u_n), \nabla T_k(u_n))\varphi'_k
 (\omega_n)\nabla (T_k(u_n) - T_k(u)) dx \\
&\quad -  \int_{\{|u_n| > k\}} a(x,T_k(u_n), 
 \nabla T_k(u_n))\varphi'_k(\omega_n)\nabla (T_k(u_n) - T_k(u)) dx.
\end{aligned}\label{aq31}
\end{equation}
Since $ 1 \leq \varphi'_k(\omega_n) \leq \varphi'_k(2k)$, it follows that
\begin{align*}
&{-\int_{\{|u_n| > k\}} a(x,T_k(u_n), \nabla T_k(u_n))
 \varphi'_k(\omega_n)\nabla (T_k(u_n) - T_k(u)) dx } \\
& = \int_{\{|u_n| > k\}} a(x,T_k(u_n), \nabla T_k(u_n))
 \varphi'_k(\omega_n)\nabla T_k(u) dx \\
&\leq \varphi'_k(2k)\int_{\{|u_n| > k\}} |a(x,T_k(u_n), 
 \nabla T_k(u_n))| |\nabla T_k(u)| dx,
\end{align*}
and since $ (|a(x,T_k(u_n), \nabla T_k(u_n))| )_n $ is bounded in
 $ L^{p'(x)}(\Omega)$, then there exists
 $ \vartheta \in L^{p'(x)}(\Omega) $ such that
$$ 
|a(x,T_k(u_n), \nabla T_k(u_n))| \rightharpoonup \vartheta \quad 
\text{in }\quad L^{p'(x)}(\Omega),
$$
then
$$ 
\int_{\{|u_n| > k\}} |a(x,T_k(u_n), \nabla T_k(u_n))| |\nabla T_k(u)| dx 
\to \int_{\{|u| > k\}} \vartheta |\nabla T_k(u)| dx = 0,
$$
and we obtain
\begin{equation}
\int_{\{|u_n| > k\}} a(x,T_k(u_n), \nabla T_k(u_n)) 
\varphi'_k(\omega_n)\nabla (T_k(u_n) - T_k(u)) dx 
= \varepsilon_0(n),\label{aqp}
\end{equation}
with $ \varepsilon_0(n) $ tend to $0$ as $ n\to\infty$.

On the other hand, for the second term  on the right hand side 
of \eqref{aq30}, taking $ z_n = u_n - T_{h}(u_n) + T_k(u_n) - T_k(u)$, 
\begin{equation}
\begin{aligned}
&\int_{\{|u_n| > k\}} a(x,T_{M}(u_n), 
 \nabla T_{M}(u_n))\varphi'_k(\omega_n)
 \nabla T_{2k}(u_n-T_{h}(u_n)+T_k(u_n) - T_k(u)) dx \\
&= \int_{\{|u_n| > k\}\cap \{|z_n|\leq 2k\}} a(x,T_{M}(u_n), 
 \nabla T_{M}(u_n))\varphi'_k(\omega_n)\nabla (u_n-T_{h}(u_n)+T_k(u_n)\\
&\quad  - T_k(u)) dx \\
&= \int_{\{|u_n| > k\}\cap \{|z_n|\leq 2k\}} a(x,T_{M}(u_n), 
 \nabla T_{M}(u_n))\varphi'_k(\omega_n)\nabla (u_n - T_k(u)).\chi_{\{|u_n|> h\}}
  dx \\
&\quad - \int_{\{|u_n| > k\}\cap \{|z_n|\leq 2k\}} a(x,T_{M}(u_n), 
 \nabla T_{M}(u_n))\varphi'_k(\omega_n)\nabla T_k(u).\chi_{\{|u_n|\leq h\}} dx \\
&\geq  - \int_{\{|u_n| > k\}} |a(x,T_{M}(u_n), 
 \nabla T_{M}(u_n))| |\nabla T_k(u)| \varphi'_k(\omega_n) dx.
\end{aligned}\label{aq32}
\end{equation}
By combining \eqref{aq30}-\eqref{aqp}  and  \eqref{aq32}, we obtain
\begin{align*}
& \int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n))\varphi'_k(\omega_n) 
\nabla \omega_n  dx \\
&\geq  \int_{\Omega} a(x,T_k(u_n), \nabla T_k(u_n))\varphi'_k(\omega_n)\nabla 
 (T_k(u_n) - T_k(u)) dx \\
&\quad - \int_{\{|u_n| > k\}} |a(x,T_{M}(u_n), \nabla T_{M}(u_n))| 
|\nabla T_k(u)| \varphi'_k(\omega_n)  dx - \varepsilon_0(n), 
\end{align*} %5.16
which is equivalent to
\begin{align*}
&{\int_{\Omega}  ( a(x,T_k(u_n), \nabla T_k(u_n)) - a(x,T_k(u_n),
  \nabla T_k(u)))(\nabla T_k(u_n) - \nabla T_k(u)) \varphi'_k(\omega_n) dx}  \\
&\leq   \int_{\{|u_n| > k\}} |a(x,T_{M}(u_n), \nabla T_{M}(u_n))| 
 |\nabla T_k(u)| \varphi'_k(\omega_n) dx \\
& \quad  + \int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n))
\varphi'_k(\omega_n)\nabla \omega_n dx \\
& \quad - \int_{\Omega}  a(x,T_k(u_n), \nabla T_k(u))(\nabla T_k(u_n) 
- \nabla T_k(u)) \varphi'_k(\omega_n)  dx + \varepsilon_0(n).
\end{align*}
We obtain
\begin{equation}
\begin{aligned}
&{\int_{\Omega} ( a(x,T_k(u_n), \nabla T_k(u_n)) - a(x,T_k(u_n), 
 \nabla T_k(u)))  \varphi'_k(\omega_n) (\nabla T_k(u_n) - \nabla T_k(u))  dx}  \\
& \leq \varphi'_k(2k)\int_{\{|u_n| > k\}} |a(x,T_{M}(u_n),
  \nabla T_{M}(u_n))| |\nabla T_k(u)| dx \\
& + \int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n)) \varphi'_k(\omega_n) 
 \nabla \omega_n dx \\
& + \varphi'_k(2k)\int_{\Omega}  |a(x,T_k(u_n), \nabla T_k(u))| 
 |\nabla T_k(u_n) - \nabla T_k(u)|  dx + \varepsilon_0(n).
\end{aligned}\label{aq33}
\end{equation}
Now, we study each terms on the right hand side of the above inequality. 
For the first term, we have $ (|a(x,T_{M}(u_n),\nabla T_{M}(u_n))|)_n $ 
is bounded in $ L^{p'(x)}(\Omega)$, and since
$$ 
|\nabla T_k(u)|^{p(x)}\chi_{\{|u_n|>k\}} \leq |\nabla T_k(u)|^{p(x)},
$$
and
$$
 |\nabla T_k(u)|^{p(x)}\chi_{\{|u_n|>k\}}  \to 0, \quad  
\text{a.e. in $\Omega$ as } n\to \infty,
$$
 by the Lebesgue dominated convergence theorem, we deduce that
$$ 
|\nabla T_k(u)|\chi_{\{|u_n|>k\}}  \to 0, \quad 
\text{in $L^{p(x)}(\Omega)$ as } n\to \infty,
$$
which implies that the first term in the right hand side of  \eqref{aq33}
tends to  $0$ as $n$ tends to $\infty$, and we can write
\begin{equation}
\varphi'_k(2k)\int_{\{|u_n| > k\}} |a(x,T_{M}(u_n),\nabla T_{M}(u_n))|
|\nabla T_k(u)| dx  = \varepsilon_1(n).\label{aq34}
\end{equation}

For the third term on the right-hand side of \eqref{aq33}, we have
$$ 
|a(x,T_k(u_n),\nabla T_k(u))| \to |a(x,T_k(u),\nabla T_k(u))| \quad 
\text{in $L^{p'(x)}(\Omega)$ as } n\to\infty,
$$
and since $\nabla T_k(u_n)$ tends weakly to  
$\nabla T_k(u)$ in $(L^{p(x)}(\Omega))^{N}$, we obtain
$$  
\varphi'_k(2k)\int_{\Omega} |a(x,T_k(u_n),\nabla T_k(u))| |\nabla T_k(u_n) 
- \nabla T_k(u)|  dx \to 0 \quad \text{as }   n\to \infty,
$$
then
\begin{equation}
 \varphi'_k(2k)\int_{\Omega} |a(x,T_k(u_n),\nabla T_k(u))| |\nabla T_k(u_n) 
- \nabla T_k(u)|  dx = \varepsilon_{2}(n).\label{aq38}
\end{equation}
 By \eqref{aq33} we conclude that
\begin{equation}
\begin{aligned}
& \int_{\Omega} (a(x,T_k(u_n),\nabla T_k(u_n)) 
- a(x,T_k(u_n),\nabla T_k(u)))\varphi'_k(\omega_n) (\nabla T_k(u_n) 
- \nabla T_k(u))  dx \\
&\leq \int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n))\varphi'_k(\omega_n)\nabla 
\omega_n dx + \varepsilon_{3}(n).
\end{aligned}\label{aq39}
\end{equation}

 Now, we turn to the second term on the left-hand side of \eqref{aq29};
by \eqref{aq5} we have 
\begin{align*}
&\big|\int_{\{|u_n|\leq k\}} g_n(x,u_n,\nabla u_n)\varphi_k(\omega_n) dx\big| \\
& \leq \int_{\{|u_n|\leq k\}} b(|u_n|)(c(x) 
 + |\nabla T_k(u_n)|^{p(x)})|\varphi_k(\omega_n)| dx \\
&\leq b(k)\int_{\{|u_n|\leq k\}} 
  c(x)|\varphi_k(\omega_n)| dx\\
&\quad  + \frac{b(k)}{\alpha}\int_{\Omega}  a(x,T_k(u_n),\nabla 
 T_k(u_n))\nabla T_k(u_n)|\varphi_k(\omega_n)| dx \\
& \leq b(k)\int_{\{|u_n|\leq k\}} c(x)|\varphi_k(\omega_n)| dx 
+ \frac{b(k)}{\alpha}\int_{\Omega} (a(x,T_k(u_n),\nabla T_k(u_n)) \\
&\quad - a(x,T_k(u_n),\nabla T_k(u)))(\nabla T_k(u_n) 
 - \nabla T_k(u))|\varphi_k(\omega_n)| dx \\
&\quad + \frac{b(k)}{\alpha}\int_{\Omega} a(x,T_k(u_n),
 \nabla T_k(u))(\nabla T_k(u_n) - \nabla T_k(u)) |\varphi_k(\omega_n)| dx \\
&\quad + \frac{b(k)}{\alpha}\int_{\Omega}  a(x,T_k(u_n),
 \nabla T_k(u_n)) \nabla T_k(u) |\varphi_k(\omega_n)| dx.
\end{align*}
Then
\begin{equation}
\begin{aligned}
&\frac{b(k)}{\alpha} \int_{\Omega} ( a(x,T_k(u_n),\nabla T_k(u_n)) 
 - a(x,T_k(u_n),\nabla T_k(u)))(\nabla T_k(u_n) \\
&- \nabla T_k(u)) |\varphi_k(\omega_n)| dx \\
& \geq \big|\int_{\{|u_n|\leq k\}} g_n(x,u_n,\nabla u_n)
 \varphi_k(\omega_n) dx\big| - b(k)
 \int_{\{|u_n|\leq k\}} c(x)|\varphi_k(\omega_n)| dx \\
&\quad - \frac{b(k)}{\alpha}\int_{\Omega} a(x,T_k(u_n),
 \nabla T_k(u))(\nabla T_k(u_n) - \nabla T_k(u)) |\varphi_k(\omega_n)| dx \\
&\quad - \frac{b(k)}{\alpha}\int_{\Omega}  a(x,T_k(u_n),\nabla T_k(u_n))
 \nabla T_k(u) |\varphi_k(\omega_n)| dx.
\end{aligned}\label{aq40}
\end{equation}
We have
\begin{equation}
\int_{\{|u_n|\leq k\}} c(x)|\varphi_k(\omega_n)| dx  
\to \int_{\{|u|\leq k\}} c(x)|\varphi_k(T_{2k}(u - T_{h}(u)))| dx  
= 0\quad\text{as } n\to \infty.
\end{equation}
Concerning the third term on the right hand side of \eqref{aq40}, we have
\begin{align*}
&\int_{\Omega} a(x,T_k(u_n),\nabla T_k(u))(\nabla T_k(u_n) 
 - \nabla T_k(u))|\varphi_k(\omega_n)| dx  \\
&\leq \varphi_k(2k)\int_{\Omega} |a(x,T_k(u_n),\nabla T_k(u))| |\nabla T_k(u_n) 
 - \nabla T_k(u)| dx,
\end{align*}
and by \eqref{aq38}, we deduce that
\begin{equation}
\int_{\Omega} a(x,T_k(u_n),\nabla T_k(u))(\nabla T_k(u_n) 
- \nabla T_k(u)) |\varphi_k(\omega_n)| dx \to  0\quad 
\text{as } n \to \infty.\label{aq41}
\end{equation}
For the last term of right hand side of  \eqref{aq40}, we have 
$ (a(x,T_k(u_n),\nabla T_k(u_n)))_n $ is bounded in $ (L^{p'(x)}(\Omega))^{N}$, 
then there exists $ \varphi \in (L^{p'(x)}(\Omega))^{N} $ such that 
\[
 a(x,T_k(u_n),\nabla T_k(u_n)) \rightharpoonup \varphi \]
 in 
$ (L^{p'(x)}(\Omega))^{N}$, and since
$$ 
\nabla T_k(u) |\varphi_k(\omega_n)| \to \nabla T_k(u) 
|\varphi_k(T_{2k}(u - T_{h}(u)))| \quad\text{in }   (L^{p(x)}(\Omega))^{N} ,
$$
it follows that
\begin{equation}
\begin{aligned}
&\int_{\Omega}  a(x,T_k(u_n),\nabla T_k(u_n))\nabla T_k(u)
|\varphi_k(\omega_n)| dx\\
& \to \int_{\Omega}  
\varphi\nabla T_k(u)|\varphi_k(T_{2k}(u - T_{h}(u)))| dx = 0.
\end{aligned}\label{aq42}
\end{equation}
Combining \eqref{aq40}, \eqref{aq41} and \eqref{aq42}, we obtain
\begin{equation}
\begin{aligned}
&\frac{b(k)}{\alpha}\int_{\Omega} ( a(x,T_k(u_n),\nabla T_k(u_n)) \\
&- a(x,T_k(u_n),\nabla T_k(u)))(\nabla T_k(u_n)
 - \nabla T_k(u)) |\varphi_k(\omega_n)| dx \\
& \geq \Big|\int_{\{|u_n|\leq k\}} g_n(x,u_n,\nabla u_n)
\varphi_k(\omega_n) dx\Big| + \varepsilon_{4}(n).
\end{aligned}\label{aq43}
\end{equation}
Thanks to  \eqref{aq39}  and  \eqref{aq43}, we obtain
\begin{equation}
\begin{aligned}
&\int_{\Omega} \big(a(x,T_k(u_n), \nabla T_k(u_n)) - a(x,T_k(u_n), 
\nabla T_k(u))\big)\\
&\quad\times\big(\nabla T_k(u_n) - \nabla T_k(u)\big) 
\Big(\varphi'_k(\omega_n) 
- \frac{b(k)}{\alpha}.|\varphi_k(\omega_n)|\Big) dx  \\
& \leq  \int_{\Omega} a(x,T_{M}(u_n), \nabla T_{M}(u_n))
 \varphi'_k(\omega_n)\nabla \omega_n dx \\
&\quad - \big|\int_{\{|u_n|\leq k\}} g_n(x,u_n,\nabla u_n)
 \varphi_k(\omega_n) dx\big|  + \varepsilon_{5}(n) \\
& \leq  \int_{\Omega} f_n\varphi_k(\omega_n)   dx 
 + \int_{\{|u_n| \leq M\}} \phi_n(T_{M}(u_n))\varphi'_k(\omega_n)\nabla \omega_n
   dx  + \varepsilon_{5}(n).
\end{aligned}\label{aq44}
\end{equation}
We have $ \omega_n\rightharpoonup T_{2k}(u-T_{h}(u)) $ 
weak-$*$ in $ L^{\infty}(\Omega) $ then
\begin{equation}
\int_{\Omega} f_n\varphi_k(\omega_n)   dx \to \int_{\Omega} 
f\varphi_k(T_{2k}(u-T_{h}(u)))  dx\quad \text{as } n\to\infty,\label{aq45}
\end{equation}
and for $n$ large enough (for example $n \geq M$), we can write
$$
\int_{\Omega} \phi_n(T_{M}(u_n))\varphi'_k(\omega_n)\nabla \omega_n  dx  
= \int_{\{|u_n|\leq M\}} \phi(T_{M}(u_n))\varphi'_k(\omega_n)\nabla \omega_n  dx,
$$
it follows that
\begin{equation}
\begin{aligned}
&\int_{\Omega} \phi_n(T_{M}(u_n))\varphi'_k(\omega_n)\nabla \omega_n  dx  \\
&\to \int_{\Omega} \phi(T_{M}(u))\varphi'_k(T_{2k}(u-T_{h}(u)))\nabla 
T_{2k}(u-T_{h}(u))  dx\quad \text{as } n\to\infty.
\end{aligned}\label{ap455}
\end{equation}
Combining \eqref{aq44} and \eqref{ap455}, we obtain
\begin{equation}
\begin{aligned}
& \frac{1}{2}\int_{\Omega} ( a(x,T_k(u_n), \nabla T_k(u_n)) 
- a(x,T_k(u_n), \nabla T_k(u)))(\nabla T_k(u_n) - \nabla T_k(u)) dx  \\
& \leq \int_{\Omega} f\varphi_k(T_{2k}(u-T_{h}(u)))  dx \\
&\quad + \int_{\Omega} \phi(T_{M}(u))\varphi'_k(T_{2k}(u-T_{h}(u)))
\nabla T_{2k}(u-T_{h}(u))  dx + \varepsilon_{6}(n).
\end{aligned}\label{aq46}
\end{equation}
Taking $ \Psi(t) = {\int_0^{t} \phi(\tau)\varphi'_k(\tau- T_{h}(\tau))  d\tau}$, 
then $ \Psi(0) = 0_{\mathbb{R}^{N}} $ and $ \Psi \in C^1(\mathbb{R}^{N})$.
By the Divergence Theorem (see also \cite{bocc2}), we obtain
\begin{align*}
& \int_{\Omega} \phi(T_{M}(u))\varphi'_k(T_{2k}(u-T_{h}(u)))
 \nabla T_{2k}(u-T_{h}(u))  dx  \\
&= \int_{\{h < |u| \leq 2k + h\}} \phi(u)\varphi'_k(u-T_{h}(u)) \nabla u   dx \\
& = \int_{\{|u| \leq 2k + h\}} \phi(T_{2k + h}(u)) \varphi'_k(T_{2k + h}(u)
 -T_{h}(u))\nabla T_{2k + h}(u)  dx \\
& \quad  -  \int_{\{|u| \leq h\}} \phi(T_{h}(u))\varphi'_k(T_{h}(u)
 - T_{h}(u)) \nabla T_{h}(u)  dx \\
& = \int_{\Omega} \operatorname{div} \Psi(T_{2k + h}(u)) dx 
-  \int_{\Omega} \operatorname{div} \Psi(T_{h}(u)) dx \\
& = \int_{\partial \Omega} \Psi(T_{2k + h}(u)).\overrightarrow{n} dx
  -  \int_{\partial \Omega} \Psi(T_{h}(u)).\overrightarrow{n} dx \\
& = \sum_{i = 1}^{N}\Big(\int_{\partial \Omega} \Psi_{i}(T_{2k + h}(u)).n_{i} dx
 -  \int_{\partial \Omega} \Psi_{i}(T_{h}(u)).n_{i} dx\Big) = 0,
\end{align*}
since $ u = 0 $ on $ \partial \Omega$, with 
$ \Psi = (\Psi_1, \ldots ,\Psi_{N}) $ and 
$ \overrightarrow{n} = (n_1,n_{2}, \ldots ,n_{N}) $ 
the normal vector on $ \partial \Omega$. Then, by letting $h$ tend
 to infinity in \eqref{aq46}, we obtain
\begin{equation}
\int_{\Omega} (a(x,T_k(u_n),\nabla T_k(u_n)) 
- a(x,T_k(u_n),\nabla T_k(u)))(\nabla T_k(u_n) 
- \nabla T_k(u))  dx \to 0 \label{aq47}
\end{equation}
as $n\to \infty$.
Using  Lemma \ref{lemp4}, we deduce that
\begin{equation}
 T_k(u_n) \to T_k(u) \quad \text{in }  W_0^{1,p(x)}(\Omega);\label{aq48}
\end{equation}
then
$$
\nabla u_n\to \nabla u \quad  \text{a.e. in }  \Omega.
$$

\subsection*{Step 4: Equi-integrability of $ g_n(x,u_n, \nabla u_n)$}
To prove that
$$ g_n(x,u_n,\nabla u_n) \to g(x,u,\nabla u) \quad \text{strongly in }
 L^1(\Omega),
$$
using Vitali’s theorem, it is sufficient to prove that 
$ g_n(x, u_n,\nabla u_n) $ is uniformly equi-integrable.
Indeed, taking $ T_1(u_n - T_{h}(u_n)) $ as a test function in  \eqref{aq9}, 
we obtain
\begin{equation}
\begin{aligned}
&\int_{\Omega} a(x,u_n, \nabla u_n)\nabla T_1(u_n - T_{h}(u_n))  dx 
 + \int_{\Omega} g_n(x,u_n,\nabla u_n)T_1(u_n - T_{h}(u_n))  dx \\
& = \int_{\Omega} f_n T_1(u_n - T_{h}(u_n))  dx  
 + \int_{\Omega} \phi_n(u_n) \nabla T_1(u_n - T_{h}(u_n))  dx,
\end{aligned}\label{aq49}
\end{equation}
which is equivalent to
\begin{equation}
\begin{aligned}
& \int_{\{h < |u_n|\leq h+ 1\}} a(x,u_n, \nabla u_n) \nabla u_n  dx 
 + \int_{\{h \leq |u_n|\}} g_n(x,u_n,\nabla u_n)T_1(u_n - T_{h}(u_n))  dx  \\
& = \int_{\{h \leq |u_n|\}} f_n T_1(u_n - T_{h}(u_n))  dx 
  + \int_{\{h < |u_n|\leq h+ 1\}} \phi_n(u_n) \nabla u_n   dx.
\end{aligned}\label{aq50}
\end{equation}
Taking $ \Phi_n(t) = {\int_0^{t} \phi_n(\tau)  d\tau}$, we have
 $ \Phi_n(0) = 0_{\mathbb{R}^{N}} $ and $ \Phi_n \in C^1(\mathbb{R}^{N})$.
In view of the Divergence theorem, 
\begin{align*}
&\int_{\{h < |u_n|\leq h+ 1\}} \phi_n(u_n)\nabla u_n dx \\
& = \int_{\{|u_n|\leq h+ 1\}} \phi_n(u_n)\nabla u_n dx
 - \int_{\{|u_n|\leq h\}} \phi_n(u_n)\nabla u_n dx \\
& = \int_{\Omega} \phi_n(T_{h+1}(u_n))\nabla T_{h+1}(u_n) dx 
 - \int_{\Omega} \phi_n(T_{h}(u_n))\nabla T_{h}(u_n) dx \\
& = \int_{\Omega} \operatorname{div} \Phi_n(T_{h+1}(u_n)) dx
 - \int_{\Omega} \operatorname{div} \Phi_n(T_{h}(u_n)) dx \\
& = \int_{\partial\Omega} \Phi_n(T_{h+1}(u_n)).\overrightarrow{n} d\sigma
 - \int_{\partial\Omega} \Phi_n(T_{h}(u_n)).\overrightarrow{n} d\sigma = 0.
\end{align*}
Since $ u_n = 0 $ on $\partial \Omega$, with 
$ \Phi_n = (\Phi_{n,1}, \ldots ,\Phi_{n,N})$, and since
$$
\int_{\{h < |u_n|\leq h+ 1\}} a(x,u_n, \nabla u_n) \nabla u_n  dx \geq 0, 
$$
it follows that
\begin{align*}
\int_{\{h + 1\leq |u_n|\}} |g_n(x,u_n,\nabla u_n)|  dx 
& {= \int_{\{h + 1 \leq |u_n|\}} g_n(x,u_n,\nabla u_n)T_1(u_n - T_{h}(u_n))  dx} \\
& {\leq  \int_{\{h \leq |u_n|\}} g_n(x,u_n,\nabla u_n)T_1(u_n - T_{h}(u_n))  dx} \\
& {\leq \int_{\{h \leq |u_n|\}} f_nT_1(u_n - T_{h}(u_n))  dx} \\
& {\leq \int_{\{h \leq |u_n|\}} |f_n|  dx,}
\end{align*}
thus, for all $ \eta > 0$, there exists $ h(\eta) > 0 $ such that
\begin{equation}
\int_{\{h(\eta) \leq |u_n|\}} |g_n(x,u_n,\nabla u_n)|  dx 
\leq  \frac{\eta}{2}.\label{aq51}
\end{equation}
On the other hand, for any measurable subset $ E\subset \Omega$, we have
\begin{equation}
\begin{aligned}
\int_{E} |g_n(x,u_n,\nabla u_n)|  dx 
&\leq \int_{E \cap \{|u_n| < h(\eta)\}} b(h(\eta))(c(x) 
+ |\nabla u_n|^{p(x)})  dx \\
&\quad + \int_{\{|u_n| \geq h(\eta)\}} |g_n(x,u_n,\nabla u_n)|  dx,
\end{aligned}\label{aq52}
\end{equation}
thanks to \eqref{aq48}, there exists $ \beta(\eta) > 0 $ such that
\begin{equation}
\int_{E \cap \{|u_n| < h(\eta)\}} b(h(\eta))(c(x) 
+ |\nabla u_n|^{p(x)})  dx \leq \frac{\eta}{2} \quad \text{for } 
 \operatorname{meas}(E) \leq \beta(\eta).\label{aq53}
\end{equation}
Finally, by combining \eqref{aq51}, \eqref{aq52} and \eqref{aq53}, we obtain
\begin{equation}
\int_{E} |g_n(x,u_n,\nabla u_n)|  dx \leq \eta, \quad \text{with }
  \operatorname{meas}(E) \leq \beta(\eta).\label{aq54}
\end{equation}
Then $ (g_n(x,u_n,\nabla u_n))_n $ is equi-integrable, and by the Vitali's 
Theorem we deduce that
\begin{equation}
g_n(x,u_n,\nabla u_n) \to g(x,u,\nabla u) \quad\text{in } L^1(\Omega).
\end{equation}

\subsubsection*{Step 5: Passage to the limit}
 Let $ \varphi\in W_0^{1,p(x)}(\Omega)\cap L^{\infty}(\Omega) $ and 
$  M = k + \|\varphi\|_{\infty} $ with $ k> 0$, we will show that
 $$ 
\liminf_{n \to \infty} \int_{\Omega} a(x,u_n,\nabla u_n) 
\nabla T_k(u_n - \varphi)  dx \geq \int_{\Omega} a(x,u,\nabla u) 
\nabla T_k(u - \varphi)  dx.
$$
 If $|u_n|>M$ then $ |u_n- \varphi|\geq |u_n|-\|\varphi\|_{\infty} > k$; 
therefore $\{|u_n- \varphi|\leq k\}\subseteq \{|u_n|\leq M\}$,  
which implies that
\begin{equation}
\begin{aligned}
&   a(x,u_n,\nabla u_n)\nabla T_k(u_n - \varphi) \\
& = a(x,u_n,\nabla u_n)\nabla (u_n - \varphi)\chi_{\{|u_n- \varphi|\leq k\}} \\
& =  a(x,T_{M}(u_n),\nabla T_{M}(u_n))(\nabla T_{M}(u_n)
 - \nabla \varphi)\chi_{\{|u_n- \varphi|\leq k\}}.
\end{aligned}\label{aq55}
\end{equation}
Then
\begin{equation}
\begin{aligned}
&\int_{\Omega}  a(x,u_n,\nabla u_n)\nabla T_k(u_n - \varphi) dx\\
& =  \int_{\Omega} a(x,T_{M}(u_n)\nabla T_{M}(u_n))(\nabla T_{M}(u_n) 
 - \nabla \varphi)\chi_{\{|u_n- \varphi|\leq k\}} dx \\
&=  \int_{\Omega} (a(x,T_{M}(u_n),\nabla T_{M}(u_n)) 
 - a(x,T_{M}(u_n),\nabla \varphi))\\
&\quad\times (\nabla T_{M}(u_n) 
 - \nabla \varphi)\chi_{\{|u_n- \varphi|\leq k\}} dx \\
&  \quad + \int_{\Omega} a(x,T_{M}(u_n),\nabla \varphi)(\nabla T_{M}(u_n) 
- \nabla \varphi)\chi_{\{|u_n- \varphi|\leq k\}} dx,
\end{aligned}\label{aq56}
\end{equation}
 we obtain
\begin{equation}
\begin{aligned}
&\liminf_{n\to +\infty} \int_{\Omega}  a(x, u_n,\nabla u_n)\nabla T_k(u_n 
 - \varphi) dx\\
&\geq \int_{\Omega} (a(x,T_{M}(u),\nabla T_{M}(u)) 
 - a(x,T_{M}(u),\nabla \varphi))(\nabla T_{M}(u)
  - \nabla \varphi)\chi_{\{|u- \varphi|\leq k\}} dx\\ 
& \quad + \lim_{n\to +\infty} \int_{\Omega} a(x,T_{M}(u_n),
 \nabla \varphi)(\nabla T_{M}(u_n) 
 - \nabla \varphi)\chi_{\{|u_n- \varphi|\leq k\}} dx.
\end{aligned}\label{aq57}
\end{equation}
Note that the second  term in the right hand side of \eqref{aq57} is equal to
$$
\int_{\Omega} a(x,T_{M}(u),\nabla \varphi)(\nabla T_{M}(u) 
- \nabla \varphi)\chi_{\{|u- \varphi|\leq k\}}dx.
$$
Finally, we have
\begin{align*}
&\liminf_{n\to +\infty} \int_{\Omega}  a(x, u_n,\nabla u_n)\nabla T_k(u_n
  - \varphi) dx  \\
&\geq \int_{\Omega} a(x,T_{M}(u),\nabla T_{M}(u))(\nabla T_{M}(u) 
 - \nabla \varphi)\chi_{\{|u- \varphi|\leq k\}} dx, \\
&  = \int_{\Omega} a(x,u,\nabla u)(\nabla u 
 - \nabla \varphi)\chi_{\{|u- \varphi|\leq k\}} dx \\
& =   \int_{ \Omega} a(x,u,\nabla u)\nabla T_k(u - \varphi) dx.
\end{align*}
Now, taking $T_k(u_n-\varphi)$ as a test function  
in \eqref{aq9} and passing to the  limit, we conclude the desired statement.
This completes the 5 steps for the proof of Theorem \ref{thm1}.

\begin{theorem}\label{thm2}
Assume that \eqref{aq1}-\eqref{aq5} and \eqref{aqq1} hold,
 $ p(.)\in C_{+}(\bar{\Omega}) $ such that
 $ 2 - \frac{1}{N} < p_{-}\leq p_{+} < N$. Then problem \eqref{aq8}
 has at least one solution $  u \in W_0^{1,q(x)}(\Omega)$ for all 
continuous functions $ q(.)\in C_{+}(\bar{\Omega}) $ such that
 $ 1< q(x) < \bar{q}(x) = \frac{N(p(x) - 1)}{N-1}$.
\end{theorem}

\begin{proof}
 Let $ (f_n)_n $ be a sequence in  $ W^{-1,p'(x)}(\Omega)\cap L^1(\Omega) $ 
such that $ f_n\to f $ in $ L^1(\Omega) $ and $ \|f_n\|_1 \leq \|f\|_1$.
we consider the approximate problem
\begin{equation}
\begin{gathered}
Au_n + g_n(x,u_n,\nabla u_n) = f_n - \operatorname{div} \phi_n(u_n)\\
u_n\in W_0^{1,p(x)}(\Omega), 
\end{gathered}\label{aq58}
\end{equation}
where $\phi_n(s)= \phi(T_n(s))$ and 
$ { g_n(x,s,\xi) = \frac{g(x,s,\xi)}{1+ \frac{1}{n}|g(x,s,\xi)|}}$.

Thanks to the first step in the proof of Theorem \ref{thm1}, there exists 
at least one weak solution $ u_n\in W_0^{1,p(x)}(\Omega) $ for this 
approximate problem.
Let $ \psi_k(t) $ be a real valued function
\begin{equation}
\psi_k(t) = \begin{cases}
 0  &\text{if }  0 \leq t \leq k, \\
t - k  &\text{if }  k < t \leq k +1, \\
 1   &\text{if }  k+1 < t , \\
-\psi_k(-t)   &\text{otherwise }, 
\end{cases}\label{aq59}
\end{equation}
and we define the sets
$$
B_0 = \{x\in \Omega: |u_n| \leq 1\},\quad
B_k = \{x\in \Omega: k < |u_n| \leq k +1\} \quad 
\text{for }k \in \mathbb{N}^{*}.
$$
Taking  $ \psi_k(u_n) $ as a test function in the approximate problem 
\eqref{aq58}, we obtain
\begin{align*}
& \int_{\Omega} a(x, u_n, \nabla u_n)\nabla \psi_k(u_n)  dx 
+ \int_{\Omega} g_n(x, u_n, \nabla u_n)\psi_k(u_n)  dx \\
& = \int_{\Omega} f_n\psi_k(u_n)  dx 
 + \int_{\Omega} \phi_n(u_n)\nabla \psi_k(u_n)  dx.
\end{align*}
Then
\begin{align*}
& \int_{B_k} a(x, u_n, \nabla u_n)\nabla u_n  dx 
 + \int_{\{|u_n|> k \}} g_n(x, u_n, \nabla u_n)\psi_k(u_n)  dx \\
& = \int_{\{|u_n|> k \}} f_n\psi_k(u_n)  dx 
 + \int_{B_k} \phi_n(u_n)\nabla u_n  dx\,.
\end{align*}
By the Divergence theorem,
\begin{equation}
\begin{aligned}
\int_{B_k} \phi_n(u_n)\nabla u_n dx 
& =  {\int_{\{|u_n|\leq k+1\}} \phi_n(u_n)\nabla u_n dx 
 - \int_{\{|u_n|\leq k\}} \phi_n(u_n)\nabla u_n dx} \\
& =  {\int_{\Omega} \phi_n(T_{k+1}(u_n))\nabla T_{k+1}(u_n) dx
  - \int_{\Omega} \phi_n(T_k(u_n))\nabla T_k(u_n) dx} \\
& =  \int_{\Omega} \operatorname{div} \Phi_n(T_{k+1}(u_n)) dx 
 - \int_{\Omega} \operatorname{div} \Phi_n(T_k(u_n))  dx = 0\,.
\end{aligned}\label{aq60}
\end{equation}
Since $ \psi_k(u_n) $ has the same sign as $ u_n$, 
 $ g_n(x, u_n, \nabla u_n)\psi_k(u_n) \geq 0 $ and we obtain
$$ 
\int_{B_k} a(x, u_n, \nabla u_n)\nabla u_n  dx 
\leq \int_{\{|u_n| > k \}} f_n\psi_k(u_n)  dx 
\leq \int_{\Omega} |f_n|  dx,
$$
using \eqref{aq2}, we deduce that
\begin{equation}
\alpha\int_{B_k} |\nabla u_n|^{p(x)}  dx \leq \|f\|_1\quad \text{for all } 
 k\geq 0.
\end{equation}
In view of the Lemma \ref{lemp5},  there exists a constant $ C $ 
that does not depend on $ n $ such that
$$ 
\big\|u_n\big\|_{1,q(x)} \leq C,
$$
for any continuous exponent $ q(\cdot)\in C_{+}(\overline{\Omega}) $
 with $ {1< q(x) < \bar{q}(x) = \frac{N(p(x) - 1)}{N-1}}$. 
By using the same steps in the proof of Theorem \ref{thm1},
 we can show that there exists a subsequence still denoted $ (u_n)_n $ 
 which converge to $ u$, then
$$
\big\|u\big\|_{1,q(x)} \leq C,
$$
where $ u $ is solution of  \ref{aq8}.
\end{proof}

\begin{theorem}\label{thm3}
Assume that \eqref{aq1}--\eqref{aq5} and  \eqref{aqq1} hold, 
$p(.)\in C_{+}(\overline{\Omega}) $ such that 
$ 2 - \frac{1}{N} < p_{-}\leq p_{+} < N$.
If $ f\log(1+ |f|)\in L^1(\Omega)$ then   \eqref{aq8}
 has at least one solution $  u \in W_0^{1,\bar{q}(x)}(\Omega) $ with 
$ {\bar{q}(x) = \frac{N(p(x) - 1)}{N-1}}$.
\end{theorem}

\begin{proof}
 Let $ (f_n)_n $ be a sequence in  $ W^{-1,p'(x)}(\Omega)\cap L^1(\Omega) $ 
such that $ f_n\to f $ in $ L^1(\Omega)$, with $ \|f_n\|_1 \leq \|f\|_1 $ 
and $ \|f_n\log(1+|f_n|)\|_1 \leq \|f\log(1+|f|)\|_1 $ 
(for example $ f_n = T_n(f) $). We consider the approximate problem
\begin{equation}
\begin{gathered}
 A u_n + g_n(x,u_n, \nabla u_n) = f_n - \operatorname{div}  \phi_n(u_n)\\
 u_n\in W_0^{1,p(x)}(\Omega), 
\end{gathered} \label{aq70}
\end{equation}
where $\phi_n(s)= \phi(T_n(s))$ and 
$ {g_n(x,s,\xi) = \frac{g(x,s,\xi)}{1+ \frac{1}{n}|g(x,s,\xi)|}}$, 
there exists at least one weak solution $ u_n\in W_0^{1,p(x)}(\Omega) $ 
for this approximate problem.

Let $ \psi_k(t) $ be defined by \eqref{aq59}, and
$$ 
S_k = \Big\{x\in \Omega, \quad k < |u_n|\Big\}
= \cup_{r=k}^{\infty} B_{r} \quad \quad \forall k\in \mathbb{N}.
$$
By using  $ \psi_k(u_n) $ as a test function in the approximate problem 
\eqref{aq70}, we obtain
\begin{equation}
\alpha \int_{B_k} |\nabla u_n|^{p(x)}  dx 
\leq \int_{S_k} |f_n|  dx\quad \text{for all } k\in \mathbb{N}.
\end{equation}
Let $ {\bar{q}(x) = \frac{N(p(x) - 1)}{N-1}}$, we have
\begin{align*}
\int_{\Omega} \frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}  dx
& = {\sum_{k=0}^{\infty} \int_{B_k} \frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}  dx} \\
& \leq {\sum_{k=0}^{\infty} \frac{1}{k+1} \int_{B_k} |\nabla u_n|^{p(x)}  dx} \\
& \leq {\frac{1}{\alpha}\sum_{k=0}^{\infty} \frac{1}{k+1} \int_{S_k} |f_n| dx} \\
& \leq {\frac{1}{\alpha}\sum_{k=0}^{\infty} \frac{1}{k+1} \sum_{s=k}^{\infty} \int_{B_{s}} |f_n| dx} \\
& = {\frac{1}{\alpha}\sum_{k=0}^{\infty} \sum_{s=k}^{\infty} \int_{B_{s}} 
|f_n|\frac{1}{k+1} dx} \\
& = {\frac{1}{\alpha}\sum_{s=0}^{\infty} \sum_{k=0}^{s} 
\int_{B_{s}} |f_n|\frac{1}{k+1}   dx}\,.
\end{align*}
Since $\sum_{k=0}^{\infty} \sum_{s=k}^{\infty} v_{s,k} 
= \sum_{s=0}^{\infty} \sum_{k=0}^{s} v_{s,k}$, the above expression 
equals
\begin{align*}
 {\frac{1}{\alpha}\sum_{s=0}^{\infty} \int_{B_{s}} |f_n|(\sum_{k=0}^{s} 
\frac{1}{k+1} )  dx} 
& \leq \frac{1}{\alpha}\sum_{s=0}^{\infty} \int_{B_{s}} |f_n|[1 
+ \log(1 + s)] dx \\
& {\leq \frac{1}{\alpha}\sum_{s=0}^{\infty} \int_{B_{s}} |f_n|[1 
+ \log(1 + |u_n|)] dx} \\
& {\leq \frac{1}{\alpha} \int_{\Omega} |f_n|[1 + \log(1 + |u_n|)] dx,}
\end{align*}
and since $  a b \leq  a \log(1+a) + e^{b} $ for all $ a, b \geq 0$, 
 we obtain
\begin{align*}
&\frac{1}{\alpha} \int_{\Omega} |f_n| [1 + \log(1 + |u_n|)] dx\\
 & =  {\frac{1}{\alpha} \int_{\Omega} |f_n| dx + \frac{1}{\alpha} 
 \int_{\Omega} |f_n| \log(1 + |u_n|) dx} \\
& \leq { \frac{1}{\alpha} \int_{\Omega} |f_n| dx 
 + \frac{1}{\alpha} \int_{\Omega} |f_n| \log(1 + |f_n|) dx 
 + \frac{1}{\alpha} \int_{\Omega} (1 + |u_n|) dx} \\
& \leq \frac{1}{\alpha}\|f\|_1 
 + \frac{1}{\alpha}\|f\log(1 + |f|)\|_1
 + \frac{1}{\alpha} \int_{\Omega} (1 + |u_n|) dx\,.
\end{align*}
In view of the Theorem \ref{thm2} we have $ u_n\in W_0^{1,q(x)}(\Omega)$;
then $ {\int_{\Omega}|u_n| dx} $ is bounded. It follows that
\begin{equation}
\int_{\Omega} \frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}  dx \leq C_1,\label{aq722}
\end{equation}
with $ C_1 $ is a constant that does not depend on $ n$.

Now, observe that $ \overline{\Omega} $ is compact, therefore,
 we can cover it with a finite number of balls 
$ (B_{i})_{i=1,\ldots,m}$, with $ B_{i} = B(x_{i},\delta)$. we denote
$$ 
p_{i-} = \min \{ p(x):x\in \overline{B_{i}\cap\Omega}\} 
\quad \text{and}\quad p_{i+} = \max \{ p(x)  : x\in \overline{B_{i}\cap\Omega}\},
$$
since $ p(\cdot) $ is a real-valued continuous function on $ \overline{\Omega}$, 
then, by taking  $ \delta > 0 $ small enough such that
\begin{equation}
\frac{(N - p_{i-})(p_{i-} - 1)^{2}}{N + p_{i-}^{2} - 2p_{i-}} + p_{i-}
 > p_{i+} \quad \text{in $B_{i}\cap\Omega$ for } i = 1,\ldots,m,
\end{equation}
and there exists a constant $  a > 0 $ such that
$$ 
\operatorname{meas}(B_{i} \cap \Omega) > a \quad \text{ for } i = 1,\ldots, m.
$$
By the Generalized H\"{o}lder inequality, we have
\begin{equation}
\begin{aligned}
&{\int_{B_{i}\cap\Omega} |\nabla u_n|^{\bar{q}(x)} dx}\\
& = {\int_{B_{i}\cap\Omega} 
\frac{|\nabla u_n|^{\bar{q}(x)}}{(1+ |u_n|)
 ^{\frac{\bar{q}(x)}{p(x)}}}(1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}} dx} \\
& = \int_{B_{i}\cap\Omega} \big(\frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}\big)
 ^{\frac{\bar{q}(x)}{p(x)}} (1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}} dx  \\
& \leq \big(\frac{N(p_{i+} - 1)}{(N-1)p_{i-}} 
 + \frac{N - p_{i-}}{(N-1)p_{i-}}\big)  \big\| 
\big(\frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}\big)
^{\frac{\bar{q}(x)}{p(x)}} \big\|_{L^{\frac{p(x)}{\overline{q}(x)}}
(B_{i}\cap\Omega)} \\
&\quad\times \big\| (1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}}
 \big\|_{L^{\frac{p(x)}{p(x) - \overline{q}(x)}}(B_{i}\cap\Omega)}
\end{aligned}\label{aq71}
\end{equation}
On the one hand, using \eqref{aq722} we have
\begin{equation}
\begin{aligned}
\big\| \big(\frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)}\big)
^{\frac{\bar{q}(x)}{p(x)}} \big\|_{L^{\frac{p(x)}{\overline{q}(x)}}
(B_{i}\cap\Omega)} 
& { \leq \Big(\int_{B_{i}\cap\Omega} \frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)} dx 
+ 1 \Big)^{\frac{(N-1)p_{i+}}{N(p_{i-} - 1)}}} \\
& { \leq \Big(\int_{\Omega} \frac{|\nabla u_n|^{p(x)}}{(1+ |u_n|)} dx 
+ 1\Big)^{\frac{(N-1)p_{+}}{N(p_{-} - 1)}}} \\
&  \leq (C_1 + 1)^{\frac{(N-1)p_{+}}{N(p_{-} - 1)}}
\end{aligned} \label{aq72}
\end{equation}
On the other hand, thanks to the Sobolev-Poincar\'{e} inequality, 
we have
\begin{align*}
 {\|u_n\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}} 
& {\leq \|u_n - \overline{u}_{n,i}\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)} 
 + \|\overline{u}_{n,i}\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}} \\
 & {\leq c \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)} 
 + \|\overline{u}_{n,i}\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}}
\end{align*}
with $ {\overline{u}_{n,i} = \frac{1}{|B_{i}\cap\Omega|}
\int_{B_{i}\cap\Omega} u_n dx}$, and since 
$$
 {\bar{q}^{*}(x) = \frac{\bar{q}(x)}{p(x) - \bar{q}(x)}
 = \frac{N(p(x) - 1)}{N-p(x)}},
$$
 we obtain
\begin{align*}
&\int_{B_{i}\cap\Omega} (1+ |u_n|)^{\frac{\bar{q}(x)}{p(x) - \bar{q}(x)}}  dx \\ 
& \leq {C_{2}\int_{B_{i}\cap\Omega} ( 1 + |u_n|^{\frac{\bar{q}(x)}{p(x) 
 - \bar{q}(x)}} ) dx} \\
& = {C_{2}\big( \operatorname{meas}(B_{i}\cap\Omega) 
 + \int_{B_{i}\cap\Omega} |u_n|^{\bar{q}^{*}(x)}  dx\big)} \\
& \leq {C_{2}\big( \operatorname{meas}(B_{i}\cap\Omega) 
 + \|u_n\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}^{\sigma_1}\big)} \\
& \leq C_{3}\big( \operatorname{meas}(B_{i}\cap\Omega) 
 + \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)}^{\sigma_1}
  + \|\overline{u}_{n,i}\|^{\sigma_1}_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}\big),
\end{align*}
with
$$ 
\sigma_1 = \begin{cases}
\frac{N(p_{i+} - 1)}{N-p_{i+}} & \text{if } 
   \|u_n\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)} > 1, \\
\frac{N(p_{i-} - 1)}{N-p_{i-}} & \text{if }  
  \|u_n\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)} \leq 1,
 \end{cases}
$$
since $ |\overline{u}_{n,i}| \leq \frac{1}{a}\int_{\Omega} |u_n| dx$,
it follows that $ \|\overline{u}_{n,i}\|_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)} $ 
is bounded and 
\begin{equation}
\begin{aligned}
&{\big\|(1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}}\big\|_{L^{\frac{p(x)}{p(x) 
- \bar{q}(x)}}(B_{i}\cap\Omega)}} \\
& \leq {\big(\int_{B_{i}\cap\Omega} (1 + |u_n|)^{\frac{\bar{q}(x)}{p(x) 
 - \bar{q}(x)}} dx\big)^{\sigma_{2}}} \\
& \leq {\big(C_{3}\big( \operatorname{meas}(B_{i}\cap\Omega) 
 + \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)}^{\sigma_1} 
 + \|\overline{u}_{n,i}\|^{\sigma_1}_{L^{\bar{q}^{*}(x)}(B_{i}\cap\Omega)}
 \big)\big)^{\sigma_{2}}}  \\
& \leq {C_{4} (1 + \|\nabla u_n\|_{L^{\bar{q}(x)}
 (B_{i}\cap\Omega)}^{\sigma_1\sigma_{2}}),}
\end{aligned} \label{aq73}
\end{equation}
and
$$ 
\sigma_{2} = \begin{cases}
\frac{N - p_{i-}}{(N - 1)p_{i-}}& \text{if }  
  \big\|(1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}}\big\|_{L^{\frac{p(x)}{p(x)
  - \bar{q}(x)}}(B_{i}\cap\Omega)} > 1, \\
\frac{N - p_{i+}}{(N - 1)p_{i+}}& \text{if }  
  \big\|(1+ |u_n|)^{\frac{\bar{q}(x)}{p(x)}}\big\|_{L^{\frac{p(x)}{p(x)
  - \bar{q}(x)}}(B_{i}\cap\Omega)} \leq 1.
 \end{cases}
$$
By combining \eqref{aq71},  \eqref{aq72} and \eqref{aq73}, we obtain
\[
\int_{B_{i}\cap\Omega} |\nabla u_n|^{\bar{q}(x)} dx 
  \leq C_{5} + C_{5} \|\nabla u_n\|_{L^{\bar{q}(x)}
(B_{i}\cap\Omega)}^{\sigma_1 \sigma_{2}}\,.
\]
Then
\begin{equation}
\begin{aligned}
&\|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)}^{\pi} 
- C_{5} \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)}^{\sigma_1 \sigma_{2}} \\
&\leq \int_{B_{i}\cap\Omega} |\nabla u_n|^{\bar{q}(x)} dx
 - C_{5} \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)}^{\sigma_1 \sigma_{2}} 
\leq C_{5},
\end{aligned}\label{aq74}
\end{equation}
with
$$ 
\pi = \begin{cases}
\bar{q}_{i-} & \text{if }  \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)} > 1, \\
\bar{q}_{i+} & \text{if }  \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)} 
\leq 1, \end{cases}
$$
and since  $ \sigma_{1 }\sigma_{2} < \bar{q}_{i-} \leq \pi $ 
in $ B_{i}\cap\Omega$, it follows that
$ \|\nabla u_n\|_{L^{\bar{q}(x)}(B_{i}\cap\Omega)} $ 
is bounded. Indeed, we have
\begin{align*}
&\frac{(N - p_{i-})(p_{i-} - 1)^{2}}{N + p_{i-}^{2} - 2p_{i-}} + p_{i-} 
> p_{i+} \\ 
& \Longleftrightarrow  \frac{Np_{i-}^{2} - Np_{i-} + N - p_{i-}}{N + p_{i-}^{2} 
 - 2p_{i-}} > p_{i+} \\
 & \Longleftrightarrow \quad (p_{i-} - 1)(N- p_{i+})p_{i-} 
 - (N - p_{i-})(p_{i+} - 1) > 0 \\
 & \Longleftrightarrow \quad \bar{q}_{i-} = \frac{N(p_{i-} - 1)}{N - 1} 
 > \frac{(N - p_{i-})}{(N - 1)p_{i-}}  \frac{N(p_{i+} - 1)}{N- p_{i+}} 
 \geq \sigma_{1 }\sigma_{2}. 
\end{align*}
We conclude that there exists some constants $ r_{i} > 0 $ such that 
$ \int_{B_{i}\cap\Omega} |\nabla u_n|^{\bar{q}(x)} dx \leq r_{i} $ 
for all $ i = 1,\ldots,m$, it follows that
\begin{equation}
\int_{\Omega} |\nabla u_n|^{\bar{q}(x)} dx
 = \sum_{i = 1}^{m} \int_{B_{i}\cap\Omega} |\nabla u_n|^{\bar{q}(x)} dx 
\leq C_{6},
\end{equation}
and by the Poincar\'{e} inequality, we obtain
$$ 
\big\|u_n\big\|_{1,\bar{q}(x)} \leq C_{7},
$$
with $ C_{7} $ is a constant that does not  depend on $ n$,
 we deduce that
$$
\big\|u\big\|_{1,\bar{q}(x)} \leq C_{7},
$$
where $ u $ is solution of \eqref{aq8}.
\end{proof}

\begin{theorem}\label{thm4}
Let $ p(.)\in C_{+}(\bar{\Omega})$.
Assume \eqref{aq1}-\eqref{aq5} hold with $ f\in W^{-1,p'(x)}(\Omega)$
and $ \phi\in C^{0}(\mathbb{R}^{N})$.
Then  $ \eqref{aq8} $ has at least one solution $  u \in W_0^{1,p(x)}(\Omega)$.
\end{theorem}

\begin{proof}
 Let $ u_n\in W_0^{1,p(x)}(\Omega) $ a weak solution of the approximate problem
\begin{equation}
\begin{gathered}
 A u_n + g_n(x,u_n, \nabla u_n) = f - \operatorname{div}  \phi_n(u_n)\\
 u_n\in W_0^{1,p(x)}(\Omega), 
\end{gathered} \label{aq75}
\end{equation}
where $\phi_n(s)= \phi(T_n(s))$ and
 $ {g_n(x,s,\xi) = \frac{g(x,s,\xi)}{1+ \frac{1}{n}|g(x,s,\xi)|}}$. 
By taking $ u_n $ as a test function in \eqref{aq75}, we obtain
$$ 
\int_{\Omega} a(x,u_n,\nabla u_n) \nabla u_n  dx 
+ \int_{\Omega} g_n(x,u_n,\nabla u_n) u_n  dx 
= \int_{\Omega} f u_n  dx + \int_{\Omega}\phi_n(u_n) \nabla u_n  dx.
$$
By the Divergence theorem, 
 $ \int_{\Omega}\phi_n(u_n)\nabla u_n  dx = 0$, and since 
 $ g_n(x,u_n,\nabla u_n) u_n \geq 0$, we obtain
\begin{align*}
\alpha\int_{\Omega} |\nabla u_n|^{p(x)}  dx 
& \leq \int_{\Omega} a(x,u_n,\nabla u_n)\nabla u_n  dx \\
& \leq (\frac{1}{p_{-}} + \frac{1}{p'_{-}})\|f\|_{-1,p'(x)}\|u_n\|_{1,p(x)},
\end{align*}
it follows that
$$
\|\nabla u_n\|_{p(x)}^{\gamma} \leq C_1\|f\|_{-1,p'(x)} \|u_n\|_{1,p(x)}
\quad \text{with }  
\gamma = \begin{cases}
p_{-} & \text{if }  \|\nabla u_n\|_{p(x)} > 1, \\
p_{+} & \text{if }  \|\nabla u_n\|_{p(x)} \leq 1, 
\end{cases}
$$
by using the Poincar\'{e} inequality, we obtain
$$
\|u_n\|_{1,p(x)}^{\gamma} \leq C_{2}\|u_n\|_{1,p(x)}.
$$
Then $\|u_n\|_{1,p(x)} \leq C_{3}$, 
with $C_{3}$ independent of $ n$, and
$$
\|u\|_{1,p(x)} \leq C_{3},
$$
where $ u $ is solution of the problem \eqref{aq8}.
\end{proof}


\begin{thebibliography}{00}
\bibitem{LAEA} L. Aharouch, E. Azroul; 
\emph{Existance and regurarity of entropy solutions for some nonlinear
 elliptic equations}, (Capri, 2004),
 Electronic Journal of Differential Equations, Conference 11, 2004, pp 1-10.

\bibitem{EAM} M. B. Benboubker, E. Azroul,  A. Barbara;
 \emph{On some quasilinear elliptic problems with nonstandard growths,} 
Electronic J. of Differential Equations, Vol. 2011 (2011), No. 62, pp. 1-16.

\bibitem{BenWit}   M. Bendahmane, P. Wittbold; 
\emph{Renormalized solutions for nonlinear elliptic equations with
variable exponents and L1 data}, Nonlinear Anal. 70 (2009) 567-583.

\bibitem{BBGGPV}  Ph. B\'{e}nilan, L. Boccardo, T. Gallou\"{e}t, R. Gariepy,
 M. Pierre, J.L. V\'{a}zquez; 
\emph{An $L^1$-theory of existence and uniqueness of solutions of nonlinear 
elliptic equations,} Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1995), 241-273.

\bibitem{bocc1} L. Boccardo;
 \emph{Some nonlinear Dirichlet problem in $L^1$ involving lower order 
terms in divergence form}, Progress in elliptic and parabolic partial
 differential equations (Capri, 1994), Pitman Res. Notes Math. Ser., 
350, p. 43-57, Longman, Harlow, 1996.

\bibitem{bocc3} L. Boccardo, T. Gallou\"{e}t; 
\emph{Nonlinear elliptic equations with right-hand side measures,} 
Comm. Partial Differential Equations 17 (1992), no. 3-4, 641-655.

\bibitem{bocc2} L. Boccardo, D. Giachetti, J. I. Dias, F. Murat; 
\emph{Existence and Regularity of Renormalized Solutions for Some 
Elliptic Problems Involving Derivations of Nonlinear terms}, 
journal of differential equations., 106, (1993), 215-237.

\bibitem{DHHR} L. Diening, P. Harjulehto, P. H\"{a}st\"{o},
 M. R{\aa}\v{z}i\v{c}ka;
 \emph{Lebesgue and Sobolev Spaces with Variable Exponents,} 
vol. 2017 of Lecture Notes in Mathematics, Springer, Heidelberg, Germany, 2011.

\bibitem{fan4} X. Fan, Y. Zhao, D. Zhao; 
\emph{Compact embedding theorems with symmetry of Strauss-Lions 
type for the space $W^{1,p(x)}(\Omega)$}, J.Math. Anal. Appl. 255(2001), 333-348.

\bibitem{Fan2} X. L. Fan, D. Zhao; 
\emph{On the generalised Orlicz-Sobolev Space $W^{k,p(x)}(\Omega)$}, 
J. Gansu Educ. College 12(1) (1998), 1-6.

\bibitem{Harjhast} P. Harjulehto, P. H\"{a}st\"{o}; 
\emph{Sobolev Inequalities for Variable Exponents Attaining the 
Values $1$ and $n$},  Publ. Mat. 52 (2008), no. 2, 347-363.

\bibitem{KBMESO} M. Kbiri Alaoui, D. Meskine, A. Souissi; 
\emph{On the limiting regularity result of some nonlinear elliptic equations,}
 Z. Anal. Anwend. 26 (2007), no. 4, 459-466.

\bibitem{lions} J. L. Lions; 
\emph{Quelques methodes de r\'{e}solution des probl\`{e}mes aux limites 
non lin\'{e}aires}, Dunod et Gauthiers-Villars, Paris 1969.

\bibitem{zhao} D. Zhao, W. J. Qiang, X. L. Fan; 
\emph{On generalized Orlicz spaces $L^{p(x)}(\Omega)$}, 
J. Gansu Sci. 9(2), 1997,  1-7.

\end{thebibliography}

\end{document}