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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 45, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/45\hfil Anisotropic equations with degenerate coercivity]
{Nonlinear anisotropic elliptic equations with variable exponents and
degenerate coercivity}

\author[H. Ayadi, F. Mokhtari \hfil EJDE-2018/45\hfilneg]
{Hocine Ayadi, Fares Mokhtari}

\address{Hocine Ayadi \newline
D\'epartement de math\'ematiques,
Universit\'e Mohamed Boudiaf-M'sila,
BP 166 M'sila 28000, Algeria. \newline
Laboratoire de M\'ecanique,
Physique et Mod\'elisation Math\'ematique (LM2M),
Universit\'e de M\'ed\'ea, M\'ed\'ea, Algeria}
\email{ayadi.hocine@yahoo.fr}

\address{Fares Mokhtari \newline
D\'epartement de Math\'ematiques et Informatique,
Universit\'e Benyoucef Benkhedda,
Alger 1, 2 Rue Didouche Mourad, Algeria.\newline
Laboratoire des EDPNL, ENS-Kouba, 
Alger, Algeria}
\email{fares\_maths@yahoo.fr}

\dedicatory{Communicated by Jes\'us Ildefonso D\'iaz}

\thanks{Submitted May 18, 2017. Published February 12, 2018.}
\subjclass[2010]{35J70, 35J60, 35B65}
\keywords{Anisotropic elliptic equations; variable exponents;
\hfill\break\indent degenerate coercivity; distributional solutions;
irregular data}

\begin{abstract}
 In this article, we prove the existence and the regularity of distributional
 solutions for a class of nonlinear anisotropic elliptic equations with
 $p_i(x)$ growth conditions, degenerate coercivity and $L^{m(\cdot)}$ data,
 with $m(\cdot)$ being small, in appropriate Lebesgue-Sobolev spaces with
 variable exponents. The obtained results extend some existing ones
 \cite{chen,croce}.
\end{abstract}

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

\section{Introduction}

We consider the problem
\begin{equation}\label{problem}
\begin{gathered}
-\sum^{N}_{i=1}D_i\big(a_i(x,u)| D_iu|^{p_i(x)-2}D_iu\big)
+|u|^{s(x)-1}u=f \quad \text{in } \Omega,\\
u=0\quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded open domain in $\mathbb{R}^N$ $(N \geq 2)$
with Lipschitz boundary $\partial \Omega$, and the right-hand side $f$
in $L^{1}(\Omega)$ (or $L^{m(\cdot)}(\Omega))$. We assume that the variable
exponents $s:\overline{\Omega}\to (0,+\infty)$,
$p_i:\overline{\Omega}\to (1,+\infty)$, $i=1,\dots ,N$ are continuous
functions such that
\begin{equation}\label{n-condition}
 1<\overline{p}(x)\leq N\quad \text{where }
 \frac{1}{\overline{p}(x)}
=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{p_i(x)},\quad \forall x\in\overline{\Omega}.
\end{equation}
Here, we suppose that $a_i: \Omega\times\mathbb{R}\to\mathbb{R}$,
$i=1,\dots ,N$ are Carath\'eodory functions such that for a.e. $x\in\Omega$,
for every $t\in\mathbb{R}$, we have
 \begin{equation}\label{a1-condition}
\frac{\alpha}{(1+| t|)^{\gamma_i(x)}}\leq a_i(x,t)\leq \beta,\quad
i=1,\dots ,N,
\end{equation}
where $\alpha, \beta$ are strictly positive real numbers and
$\gamma_i\in C(\overline{\Omega})$, $\gamma_i(x)\geq0$ for all
$x\in\overline{\Omega}$ and $i=1,\dots ,N$.

The main difficulty in dealing with problem \eqref{problem} is the fact that, because
of assumption \eqref{a1-condition}, the differential operator
$$
A(u)=-\sum^{N}_{i=1}D_i\big(a_i(x,u)| D_iu|^{p_i(x)-2}D_iu\big)
$$
is well defined between $\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$
 and its dual but it fails to be coercive if $u$ is large (see \cite{po}).
This shows that the classical methods for elliptic operators can't be applied.
To overcome this problem, we will proceed by approximation by means of truncatures
in $a(x,t)$ to get a coercive differential operator. We cite some papers that
 have dealt with the equation \eqref{problem} or similar problems,
see \cite{Al,bks,boc,bb,bdo,chen,f3,croce,zh,xi} and the references therein.
In case of a constant exponent $p_i(x) =2$, $s(x)=q$ and $\gamma_i(x)=\gamma$
(resp. $p_i(x) =p$) similar results can be found in \cite{chen,croce}.
The problem was also considered in \cite{zh} when $p_i(x)=p(x)$ and
$\gamma_i(x)=\gamma(x)\geq 0$, where the authors supposed that
$a_i(x,u)=a(x,u)\leq \beta(|u|)$ with $\beta : [0,+\infty) \to(0,+\infty)$
is a continuous function. The lack of growth condition on $a(x, u)$
prompted them to consider only the renormalized and entropy solutions.
 The corresponding results in the isotropic case and without lower order
term are developed in \cite{Al,boc, bb,bdo,xi}. More general results are
obtained in \cite{Al} in the constant case $p_i(x) =p$,
$\gamma_i(x)=\theta(p-1)$, $\theta\in[0,1]$. In the case $p_i(x) =p(x)$ and
$\gamma_i(x)=\theta(p(x)-1)$ where $0\leq\theta\leq \frac{p^--1}{p^+-1}$
the main results are collected in the paper \cite{xi}.
Recently, the mathematical researchers paid attention to the anisotropic
 nonlinear problems with variable exponents.
For instance, Problem \eqref{problem} was investigated in \cite{bks,mo2, mo,mok}
 under uniform ellipticity condition i.e $\gamma_i(x)=0$.
In this article we assume that the condition \eqref{a1-condition} holds
 and $f\in L^{m(\cdot)}(\Omega)$ where $p_i$ is assumed to be merely a
continuous function, and we treat the regularity of distributional
solution $u$ depending simultaneously on $s(\cdot)$ and $m(\cdot)$.

\section{Preliminaries}

 In this section we recall some facts on anisotropic spaces with variable
exponents and we give some of their properties. For further details on
the Lebesgue-Sobolev spaces with variable exponents, we refer
to \cite{aton,ld,fan1} and references therein.
In this article we set
$$
C_{+}(\overline{\Omega})=\{p\in C(\overline{\Omega}): p(x)>1,
\text{ for any $x$ in } \overline{\Omega}\}.
$$
For any $p\in C_{+}(\overline{\Omega})$, we denote
$$
p^{+}=\max_{x\in\overline{\Omega}}p(x)\quad \text{and}\quad
p^{-}=\min_{x\in\overline{\Omega}}p(x).
$$
We define the Lebesgue space with variable exponent $L^{p(\cdot)}(\Omega)$
as the set of all measurable functions $u :\Omega\to\mathbb{R}$ for which the convex modular
$$
\rho_{p(\cdot)}(u)=\int_{\Omega}|u|^{p(x)}\,dx,
$$
is finite. The expression
$$
\|u\|_{p(\cdot)}:=\|u\|_{L^{p(\cdot)}(\Omega)}
=\inf\big\{\lambda>0: \rho_{p(\cdot)}\big(\frac{u}{\lambda}\big)\leq 1\big\},
$$
defines a norm on $L^{p(\cdot)}(\Omega)$, called the Luxemburg norm.
The space $(L^{p(\cdot)}(\Omega), \|u\|_{p(\cdot)})$
is a separable Banach space. Moreover, if $1 < p^{-}\leq p^{+} < +\infty$,
then $L^{p(\cdot)}(\Omega)$ is uniformly convex, hence reflexive
and its dual space is isomorphic to $L^{p'(\cdot)}(\Omega)$
where $\frac{1}{p(x)}+\frac{1}{p'(x)}=1$. For all
$u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$,
the H\"{o}lder type inequality
$$
\big|\int_{\Omega}uv\,dx\big|
\leq \Big(\frac{1}{p^{-}}+\frac{1}{p'^{-}}\Big)\|u\|_{p(\cdot)}\|v\|_{p'(\cdot)}
\leq 2\|u\|_{p(\cdot)}\|v\|_{p'(\cdot)},
$$
holds. We define also the Banach space
$$
W^{1,p(\cdot)}(\Omega) = \big\{u \in L^{p(\cdot)}(\Omega): 
 |\nabla u|\in L^{p(\cdot)}(\Omega)\big\},
$$
which is equipped with the norm
 $$
\|u\|_{1,p(\cdot)}=\|u\|_{W^{1,p(\cdot)}(\Omega)}
=\|u\|_{p(\cdot)}+\|\nabla u\|_{p(\cdot)}.
$$
The space $(W^{1,p(\cdot)}(\Omega),\|u\|_{1,p(\cdot)})$ is a Banach space.
Next, we define $W_0^{1,p(\cdot)}(\Omega)$ the Sobolev space with
zero boundary values by
$$
W_0^{1,p(\cdot)}(\Omega)=\big\{u \in W^{1,p(\cdot)}(\Omega): u=0 \text{ on }
\partial\Omega \big\},
$$
endowed with the norm $\|\cdot\|_{1,p(\cdot)}$.
The space $W_0^{1,p(\cdot)}(\Omega)$ is separable and reflexive provided
that $1 < p^{-}\leq p^{+} < +\infty$. For $u\in W_0^{1,p(\cdot)}(\Omega)$
with $p\in C_{+}(\overline{\Omega})$, the Poincar\'e inequality holds
\begin{equation}\label{poincare}
 \|u\|_{p(\cdot)}\leq C\|\nabla u\|_{p(\cdot)},
\end{equation}
for some $C>0$ which depends on $\Omega$ and $p(\cdot)$.
Therefore, $\|\nabla u\|_{p(\cdot)}$ and $\|u\|_{1,p(\cdot)}$ are equivalent norms.

An important role in manipulating the generalized Lebesgue and Sobolev spaces
is played by the modular $\rho_{p(\cdot)}(u)$ of the space $L^{p(\cdot)}(\Omega)$.
We have the following results.

\begin{proposition}[\cite{aton,ld}]\label{pro1}
If $u_n, u\in L^{p(\cdot)}(\Omega)$ and $p^{+} < +\infty$, then the
following properties hold:
\begin{itemize}
 \item $\|u\|_{p(\cdot)}<1$ (resp. $=1, >1$)
$\Longleftrightarrow \rho_{p(\cdot)}(u)<1$ (resp. $=1, >1$),

 \item $\min\big(\rho_{p(\cdot)}(u)^{\frac{1}{p^{+}}},
\rho_{p(\cdot)}(u)^{\frac{1}{p^{-}}}\big)\leq\|u\|_{p(\cdot)}
 \leq\max\big(\rho_{p(\cdot)}(u)^{\frac{1}{p^{+}}},
\rho_{p(\cdot)}(u)^{\frac{1}{p^{-}}}\big)$,

 \item $\min\big(\|u\|_{p(\cdot)}^{p^{-}},\|u\|_{p(\cdot)}^{p^{+}}\big)
\leq\rho_{p(\cdot)}(u)
 \leq\max\Big(\|u\|_{p(\cdot)}^{p^{-}},\|u\|_{p(\cdot)}^{p^{+}}\Big)$,

 \item $\|u\|_{p(\cdot)}\leq\rho_{p(\cdot)}(u)+1$,

 \item $ \|u_n-u\|_{p(\cdot)}\to0\Longleftrightarrow\rho_{p(\cdot)}(u_n-u)\to0$.
\end{itemize}
\end{proposition}

\begin{remark} \label{rmk2.2} \rm
Note that the inequality
$$
\int_{\Omega}|f|^{p(x)}\,dx\leq C\int_{\Omega}|Df|^{p(x)}\,dx,
$$
in general does not hold (see \cite{fan}). But by Proposition \ref{pro1}
and \eqref{poincare} we have
\begin{equation}\label{ineq}
 \int_{\Omega}|f|^{p(x)}\,dx
\leq C\max\{\|Df\|_{p(\cdot)}^{p^{+}},\|Df\|_{p(\cdot)}^{p^{-}}\}.
\end{equation}
\end{remark}

Now, we present the anisotropic Sobolev space with variable exponent which is used
for the study of problem \eqref{problem}. First of all, let
$p_i(\cdot):\overline{\Omega}\to [1,+\infty)$, $i=1,\dots ,N$ be continuous
functions, we set $\vec{p}(\cdot) = (p_1(\cdot),\dots , p_{N}(\cdot))$
and $p_{+}(x)= \max_{1\leq i\leq N} p_i(x)$, for all $x\in \overline{\Omega}$.
The anisotropic variable exponent Sobolev space
$W^{1,\vec{p} (\cdot)}(\Omega)$ is defined as
$$
W^{1,\vec{p}(\cdot)}(\Omega)
=\big\{ u\in L^{p_{+}(\cdot)}(\Omega): D_iu\in L^{p_i(\cdot)}(\Omega), \;
 i=1,\dots ,N\big\},
$$
which is Banach space with respect to the norm
$$
\| u\|_{W^{1,\vec{p}(\cdot)}(\Omega)}
=\| u\|_{p_{+}(\cdot)}+\sum_{i=1}^{N}\| D_iu\|_{p_i(\cdot)}.
$$
We denote by $W_0^{1,\vec{p}(\cdot)}(\Omega)$ the closure of
$C_0^{\infty}(\Omega)$ in $W^{1,\vec{p}(\cdot)}(\Omega)$, and
we define \[
\mathring{W}^{1,\vec{p}(\cdot)}(\Omega) =W^{1,\vec{p}(\cdot)}(\Omega)\cap W_0^{1,1}(\Omega).
\]
If $\Omega$ is a bounded open set with Lipschitz boundary $\partial \Omega$,
then
$$
\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)
=\big\{u\in W^{1,\vec{p}(\cdot)}(\Omega): u|_{\partial \Omega}=0 \big\}.
$$
It is well-known that in the constant exponent case, that is, when
$\vec{p}(\cdot)=\vec{p}\in[1,+\infty)^{N}$,
$W_0^{1,\vec{p}}(\Omega)=\mathring{W}^{1,\vec{p}}(\Omega)$.
However in the variable exponent case, in general
$W_0^{1,\vec{p}(\cdot)}(\Omega)\subset
\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$
and the smooth functions are in general not dense in
$\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$, but if for each
$i=1,\dots ,N,\ p_i$ is log-H\"{o}lder continuous, that is,
there exists a positive constant $L$ such that
\begin{equation*}
 |p_i(x)-p_i(y)|\leq\frac{L}{-\ln|x-y|},\quad \forall x,y\in\Omega, \;
 |x-y|<1.
\end{equation*}
 Then $C_0^{\infty}(\Omega)$ is dense in
$\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$, thus
$W_0^{1,\vec{p}(\cdot)}(\Omega)
=\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$.
We set for all $x\in \overline{\Omega}$
$$
\overline{p}(x)=\frac{N}{\sum_{i=1}^{N}\frac{1}{p_i(x)}},
$$
and we define
$$
\overline{p}^{\star}(x)=\begin{cases}
 \frac{N\overline{p}(x)}{N-\overline{p}(x)},
&\text{for } \overline{p}(x)<N,\\
 +\infty , & \text{for } \overline{p}(x)\geq N.
\end{cases}
$$
We have the following embedding results.

\begin{lemma}[\cite{fan1}]
Let $\Omega\subset \mathbb{R}^{N}$ be a bounded domain, and
$\vec{p}(\cdot)\in(C_{+}
(\overline{\Omega}))^{N}$. If $q\in C_{+}(\overline{\Omega})$ and
for all $x\in\overline{\Omega}$, $q(x) < \max(p_{+}(x)$,
$\overline{p}^{\star}(x))$.
Then the embedding
$$
\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega),
$$
 is compact.
\end{lemma}

\begin{lemma}[\cite{fan1}]
Let $\Omega\subset \mathbb{R}^{N}$ be a  bounded domain, and
$\vec{p}(\cdot)\in(C_{+} (\overline{\Omega}))^{N}$. Suppose that
\begin{equation}\label{pcon}
 \forall x\in\overline{\Omega},\; p_{+}(x)< \overline{p}^{\star}(x).
\end{equation}
Then the following Poincar\'e-type inequality holds
\begin{equation}\label{type}
 \|u\|_{L^{p_{+}(\cdot)}(\Omega)}\leq C\sum_{i=1}^{N}\|D_iu\|_{L^{p_i(\cdot)}
(\Omega)}, \quad \forall u \in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega),
\end{equation}
where $C$ is a positive constant independent of $u$.
Thus $\sum_{i=1}^{N}\|D_iu\|_{L^{p_i(\cdot)}(\Omega)}$ is an equivalent norm
on $\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$.
\end{lemma}

The following embedding results for the anisotropic constant exponent
Sobolev space are well-known \cite{rak,troisi}.

\begin{lemma}\label{troisi-1}
Let $\alpha_i\geq1,\ i=1,\dots ,N$, we pose
$\vec{\alpha}= (\alpha_1, \dots , \alpha_{N}) $.
Suppose $u\in W^{1,\vec{\alpha}}_0(\Omega)$, and set
$$
\frac{1}{\overline{\alpha}}=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\alpha_i}, \quad
 r=\begin{cases}
 \overline{\alpha}^{\star}=\frac{N\overline{\alpha}}{N-\overline{\alpha}}
 & \text{if } \overline{\alpha}<N, \\
 \text{any number in } [1,+\infty)
& \text{if } \overline{\alpha}\geq N.
 \end{cases}
$$
Then, there exists a constant $C$ depending on $N, p_1,\dots ,p_{N}$
if $\overline{\alpha}<N$ and also on $r$ and $|\Omega|$ if
$\overline{\alpha}\geq N$, such that
\begin{equation}\label{anisotropic-inequality-1}
\| u\|_{L^{r}(\Omega)}\leq C\prod_{i=1}^{N}\| D_iu\|^{1/N}_{L^{\alpha_i}(\Omega)}.
\end{equation}
\end{lemma}

\begin{lemma}\label{troisi-1b}
Let $Q$ be a cube of $\mathbb{R}^{N}$ with faces parallel to the coordinate planes
and $\alpha_i\geq1,\ i=1,\dots ,N$. Suppose $u\in W^{1,\vec{\alpha}}(Q)$,
and set
\begin{gather*}
 r=\overline{\alpha}^{\star} \quad \text{if } \overline{\alpha}<N, \\
 r\in [1,+\infty) \quad \text{if } \overline{\alpha}\geq N.
 \end{gather*}
Then, there exists a constant $C$ depending on $N, \alpha_1,\dots ,\alpha_{N}$
if $\overline{\alpha}<N$ and also on $r$ and $|Q|$ if $\overline{r}\geq N$,
such that
\begin{equation}\label{anisotropic-inequality-2}
\| u\|_{L^{r}(Q)}\leq C\prod_{i=1}^{N}
\left(\| u\|_{L^{\alpha_i}(Q)}+\| D_iu\|_{L^{\alpha_i}(Q)}\right)^{1/N}.
\end{equation}
\end{lemma}

We will use through the paper, the truncation function $T_k$ at height
$k$ $(k > 0)$, that is $T_k(t)=\max\{-k,\min\{k,t\}\}$.

\begin{proposition}
If $u :\Omega\to \mathbb{R}$ is a measurable function such that
$T_k(u)\in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ for all $k > 0$, then
there exists a unique measurable function $v :\Omega\to \mathbb{R}^{N}$ such that
\begin{equation}\label{tancation-derivation}
 \nabla T_k(u)=v\chi_{\{| u|\leq k\}}\quad \text{a.e. in } \Omega,
\end{equation}
where $\chi_A$ denotes the characteristic function of a measurable set $A$.
Moreover, if $u\in W^{1,1}_0(\Omega)$ then $v$ coincides with the standard
distributional gradient of $u$.
\end{proposition}

A function $u$ such that $T_k(u)\in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$
for any $k > 0$, does not necessarily belong
to $W^{1,1}_0(\Omega)$. However, according to the above proposition,
it is possible to define its weak
gradient, still denoted by $\nabla u$, as the unique function $v$ which satisfies
 \eqref{tancation-derivation}.

\begin{definition} \rm
For $0<r<+\infty$, the set of all measurable functions $v: \Omega \to \mathbb{R}$
such that the functional $[u]_r=\sup_{k>0}k\operatorname{meas}
\{x\in\Omega: |u(x)|>k\}^{1/r}$ is finite is called a Marcinkiewicz
space and is denoted by $M^{r}(\Omega)$.
If $| \Omega| <\infty$ and $0 < \epsilon < r-1$, we can show that
$L^{r}(\Omega) \subset M^{r}(\Omega) \subset L^{r-\epsilon}(\Omega)$.
\end{definition}

\section{Statement of results}

\begin{definition} \rm
We say that $u$ is a distributional solution for problem \eqref{problem} if
$u \in W_0^{1,1}(\Omega)$, $| u|^{s(\cdot)}\in L^{1}(\Omega)$ and
\begin{equation}\label{solution-definition}
\sum^{N}_{i=1}\int_{\Omega} a_i(x,u)| D_i u|^{p_i(x)-2}D_iu D_i\varphi \,dx
+\int_{\Omega}|u|^{s(x)-1}u\varphi \,dx=\int_{\Omega}f \varphi \,dx,
\end{equation}
for every $\varphi \in C_0^{\infty}(\Omega)$.
\end{definition}

Our main results are the following

\begin{theorem}\label{thm1}
Let $f\in L^{1}(\Omega)$, $p_i:\overline{\Omega}\to(1,+\infty)$,
 $s:\overline{\Omega}\to(0,+\infty)$ and $\gamma_i:\overline{\Omega}\to[0,+\infty)$
be continuous functions such that for all $x\in \overline{\Omega}$,
\begin{gather}\label{cond-th2-s}
 s(x)\geq p_i(x),\quad i=1,\dots ,N, \\
\label{cond-th2-p}
\frac{\overline{p}(x)(N-1-\gamma^{+}_{+})}{N(\overline{p}(x)-1-\gamma^{+}_{+})}
<p_i(x)<k(x),
\end{gather}
 where
 $$
 k(x)=\begin{cases}
 \frac{\overline{p}(x)(N-1-\gamma^{+}_{+})}{(1+\gamma^{+}_{+})
(N-\overline{p}(x))} , & \text{if } \overline{p}(x)<N\\
 +\infty, & \text{if }\overline{p}(x)=N
 \end{cases}
 \quad\text{and}\quad
 \gamma^{+}_{+}=\max_{1\leq i\leq N}\max_{x\in\overline{\Omega}}\gamma_i(x).
 $$
Let $a_i$ be Carath\'eodory functions for $i=1,\cdots,N$ satisfying
\eqref{a1-condition}. Then, problem \eqref{problem} has at least one
distributional solution $u\in \mathring{W}^{1,\vec{q}(\cdot)}(\Omega)$
 where $q_i(\cdot)$ are continuous functions on $\overline{\Omega}$ satisfying
 \begin{equation}\label{regularite-th2}
 1\leq q_i(x)<\frac{N(\overline{p}(x)-1-\gamma^{+}_{+})p_i(x)}{\overline{p}(x)
(N-1-\gamma^{+}_{+})},\quad \forall x\in\overline{\Omega},\; i=1,\dots ,N.
 \end{equation}
\end{theorem}

\begin{theorem}\label{thm2}
 Let $f\in L^{1}(\Omega)$, $p_i:\overline{\Omega}\to(1,+\infty)$,
 $s:\overline{\Omega}\to(0,+\infty)$ and $\gamma_i:\overline{\Omega}\to[0,+\infty)$
be continuous functions such that \eqref{pcon} holds and for all
$x\in \overline{\Omega}$,
\begin{equation}\label{s-cond-th1}
 s(x)>\max\Big(\frac{1+\gamma_i(x)}{p_i(x)-1};(1+\gamma_i(x))(p_i(x)-1)\Big),
\quad i=1,\dots ,N.
\end{equation}
Let $a_i$ be Carath\'eodory functions satisfying \eqref{a1-condition}.
Then, problem \eqref{problem} has at least one distributional solution
$u\in \mathring{W}^{1,\vec{q}(\cdot)}(\Omega)\cap L^{s(\cdot)}(\Omega)$ where
$q_i(\cdot)$ are continuous functions on $\overline{\Omega}$ satisfying
\begin{equation}\label{regularite-1}
 1<q_i(x)<\frac{p_i(x)s(x)}{s(x)+1+\gamma_i(x)},\quad
 \forall x\in\overline{\Omega},\; i=1,\dots ,N.
\end{equation}
\end{theorem}

\begin{theorem}\label{thm3}
Let $m:\overline{\Omega}\to(1,+\infty)$, $p_i:\overline{\Omega}\to(1,+\infty)$,
$\gamma_i:\overline{\Omega}\to[0,+\infty)$ and
$s:\overline{\Omega}\to(0,+\infty)$ be continuous functions such that \eqref{pcon}
holds and for all $x\in \overline{\Omega}$
\begin{equation}\label{m-th-3}
 1<m(x)<h(x),\quad \nabla m\in L^{\infty}(\Omega),
\end{equation}
where
$$
h(x)=\begin{cases}
 \frac{N\overline{p}(x)}{N\overline{p}(x)+\overline{p}(x)-N},
 & \text{if }\overline{p}(x)<N \\
 \frac{p_+(x)}{p_+(x)-1}, & \text{if }\overline{p}(x)=N,
 \end{cases}
$$
and
\begin{equation}\label{s-th-3}
 s(x)\geq\frac{1+\gamma_{+}(x)}{m(x)-1}, \quad
 \nabla s\in L^{\infty}(\Omega),\quad
 \nabla \gamma_{+} \in L^{\infty}(\Omega),\quad
\gamma_{+}(x)=\max_{1\leq i\leq N}\gamma_i(x).
\end{equation}
Let $f\in L^{m(\cdot)}(\Omega)$ and let $a_i$ be Carath\'eodory functions 
satisfying  \eqref{a1-condition}. Then, problem \eqref{problem} has at least one distributional 
solution $u\in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)\cap L^{s(\cdot)m(\cdot)}(\Omega)$.
\end{theorem}

\begin{theorem}\label{thm4}
Let $f\in L^{m(\cdot)}(\Omega)$ with $m$ as in \eqref{m-th-3}, 
$p_i:\overline{\Omega}\to(1,+\infty)$, 
$\gamma_i:\overline{\Omega}\to[0,+\infty)$, and
$s:\overline{\Omega}\to(0,+\infty)$ be continuous functions. 
Assume  \eqref{pcon}, and for all $x\in \overline{\Omega}$,
\begin{equation}\label{s-th-4}
 \frac{1+\gamma_{+}(x)}{m(x)-1} >s(x)
>\max\Big(\frac{1+\gamma_i(x)}{p_i(x)m(x)-1};(1+\gamma_i(x))(p_i(x)-1)\Big), 
\end{equation}
$\nabla s\in L^{\infty}(\Omega)$,
and $i=1,\dots ,N$, where $\gamma_{+}(\cdot)=\max_{1\leq i\leq N}\gamma_i(\cdot)$.
Let $a_i$ be Carath\'eodory functions satisfying \eqref{a1-condition}. 
Then, problem \eqref{problem} has at least one distributional solution $u$ 
such that $|u|^{m(x)s(x)}\in L^{1}(\Omega)$ and $u\in \mathring{W}^{1,\vec{q}(\cdot)}(\Omega)$
where $q_i(\cdot)$ are continuous functions on $\overline{\Omega}$ satisfying
\begin{equation}\label{regularite-th-4}
 1<q_i(x)=\frac{p_i(x)m(x)s(x)}{s(x)+1+\gamma_i(x)},\quad
 \forall x\in\overline{\Omega},\; i=1,\dots ,N.
\end{equation}
\end{theorem}

\begin{remark} \label{rmk3.6} \rm
 In Theorem \ref{thm1}, it is clear that the conditions
\eqref{n-condition} and \eqref{cond-th2-p} imply that \eqref{pcon} holds 
since we have
 $$
k(x)\leq\overline{p}^{\star}(x),\ \forall x\in\overline{\Omega}.
$$
\end{remark}

\begin{remark}\label{rem} \rm
 Observe that the conditions \eqref{m-th-3}, \eqref{s-th-3}, and \eqref{pcon} 
guarantee that
 $$
s(x)>(1+\gamma_{+}(x))(p_i(x)-1),\quad \forall x\in\overline{\Omega},\ i=1,\dots ,N.
$$
\end{remark}

\begin{remark} \label{rmk3.8} \rm
 In Theorem \ref{thm4}, the conditions \eqref{m-th-3} and \eqref{pcon}
imply that the assumption \eqref{s-th-4} is not empty since we have
 \begin{equation}\label{s-remark}
 \frac{1}{m(x)-1}>p_i(x)-1,\quad \forall x\in\overline{\Omega},\; i=1,\dots ,N.
 \end{equation}
\end{remark}

\begin{remark} \label{rmk3.9} \rm
Let $f\in L^{1}(\Omega)$. Assume that for all 
$x\in\overline{\Omega}$, $\overline{p}(x)<N$ and 
$s(x)>\frac{(1+\gamma_i(x))N(\overline{p}(x)-1-\gamma^{+}_{+})}{(1+\gamma^{+}_{+})
(N-\overline{p}(x))}$ for all $i=1,\dots ,N$. Then,  assumption \eqref{cond-th2-p} 
implies \eqref{s-cond-th1} and
$$
\frac{p_i(x)s(x)}{s(x)+1+\gamma_i(x)}>\frac{N(\overline{p}(x)
-1-\gamma^{+}_{+})p_i(x)}{\overline{p}(x)(N-1-\gamma^{+}_{+})},\quad
 \forall x\in\overline{\Omega},\; i=1,\dots ,N, 
$$
so Theorem \ref{thm2} improves Theorem \ref{thm1}
(and \cite[Theorem 3.1]{bks}).
\end{remark}

\section{Approximate equation}\label{sec4}

We will use the following approximating problem
\begin{equation}\label{approximate-equation}
\begin{gathered}
-\sum^{N}_{i=1}D_i\Big(a_i(x,T_n(u_n))| D_i u_n|^{p_i(x)-2}D_iu_n\Big)
+| u_n|^{s(x)-1}u_n=T_n(f) \quad \text{in } \Omega,\\
u_n=0\quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
We are going to prove the existence of solution $u_n$ to problem
 \eqref{approximate-equation}.

\begin{lemma}\label{lemma-1}
Let $f\in L^{1}(\Omega)$ and let $s:\overline{\Omega}\to(0,+\infty)$,
$p_i:\overline{\Omega}\to(1,+\infty)$, $i = 1,\dots , N$ be continuous 
functions. Assume that \eqref{pcon} holds. Then, there exists at least 
one solution $u_n \in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ to problem
\eqref{approximate-equation} in the sense that
\begin{equation}\label{app-variationel}
\begin{aligned}
&\sum^{N}_{i=1}\int_{\Omega} a_i(x,T_n(u_n))| D_iu_n|^{p_i(x)-2}
 D_iu_nD_i\varphi \,dx
 +\int_{\Omega}| u_n|^{s(x)-1}u_n\varphi \,dx \\
&=\int_{\Omega}T_n(f) \varphi \,dx,
\end{aligned}
\end{equation}
for every $\varphi \in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$.
 Moreover
\begin{equation}\label{est1}
\int_{\Omega}| u_n|^{s(x)}\,dx\leq \int_{\Omega}| f| \,dx.
\end{equation}
\end{lemma}

\begin{proof}
Consider the  problem
\begin{equation}\label{11}
\begin{gathered}
\begin{aligned}
&-\sum^{N}_{i=1}D_i\Big(a_i(x,T_n(u_{n_k}))| D_iu_{n_k}|^{p_i(x)-2}
 D_iu_{n_k}\Big)+T_k\big(| u_{n_k}|^{s(x)-1}u_{n_k}\big)\\
&=T_n(f) \quad\text{in } \Omega,
\end{aligned}\\
u_{n_k}=0\quad  \text{on } \partial \Omega.
\end{gathered}
\end{equation}
It has been proved in \cite{fan1} that there exists a solution 
$u_{n_k} \in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ to problem \eqref{11},
 which satisfies
\begin{equation} \label{12}
\begin{aligned}
& \sum^{N}_{i=1}\int_{\Omega} a_i(x,T_n(u_{n_k}))| D_iu_{n_k}|^{p_i(x)-2}
 D_iu_{n_k}D_i\varphi \,dx \\
& +\int_{\Omega}T_k\big(| u_{n_k}|^{s(x)-1}u_{n_k}\big)\varphi \,dx\\
 &=\int_{\Omega}T_n(f) \varphi \,dx,\quad \forall
\varphi\in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega).
\end{aligned}
\end{equation}
Choosing $\varphi = u_{n_k}$ in \eqref{12}, by \eqref{a1-condition}
and using that $T_k(| u_{n_k}|^{s(x)-1})u_{n_k}\geq 0$,
we have
\begin{equation*}
\frac{\alpha}{n(1+n)^{\gamma^{+}_{+}}}\sum^{N}_{i=1}\int_{\Omega} 
| D_iu_{n_k}|^{p_i(x)}\,dx\leq \int_{\Omega} | u_{n_k}| \,dx.
\end{equation*}
Using Young's inequality for all $\varepsilon>0$, we obtain
\begin{align*}
\sum^{N}_{i=1}\int_{\Omega} | D_iu_{n_k}|^{p_i(x)}\,dx
&\leq \varepsilon{\int_{\Omega}} | u_{n_k}|^{p_{-}^{-}} \,dx+C_1 \\
&\leq \varepsilon C_2 {\int_{\Omega} }| D_i u_{n_k}|^{p_{-}^{-}} \,dx+C_1 \\
&\leq \varepsilon C_2 \sum_{i=1}^N\int_{\Omega} | D_i u_{n_k}|^{p_i(x)} \,dx+C_{3},
\end{align*}
where $C_1$, $C_2$, and $C_3$ are positive constants not depending on $k$.
 Now, we choose $\varepsilon=1/(2C_2)$, then
$$
\sum^{N}_{i=1}\int_{\Omega} | D_iu_{n_k}|^{p_i(x)}\,dx \leq C(n).
$$
It follows that the sequence $\{u_{n_k}\}_k$ is bounded in
$\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$. So, there exists a function 
$u_n\in \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ and a subsequence
(still denoted by $ u_{n_k}$) such that
\begin{equation}\label{e1}
u_{n_k}\rightharpoonup u_n \quad \text{weakly in }
 \mathring{W}^{1,\vec{p}(\cdot)}(\Omega) \text{ and a.e in } \Omega.
\end{equation}
Choosing $\varphi= u_{n_k}- u_n$ in \eqref{12} as a test function,
we can easily prove that, for all $i=1,\dots ,N$,
\[
\int_{\Omega} a_i(x,T_n(u_{n_k}))\left[| D_i u_{n_k}|^{p_i(x)-2}
D_i u_{n_k}-| D_i u_n|^{p_i(x)-2}D_i u_n\right] D_i(u_{n_k}
- u_n) \,dx
\to 0 
\]
as $k\to +\infty$.
By \eqref{a1-condition}, we obtain
\[
E_i(k)=\int_{\Omega}\big[| D_i u_{n_k}|^{p_i(x)-2}D_i u_{n_k}
-| D_i u_n|^{p_i(x)-2}D_i u_n\big] D_i(u_{n_k}- u_n) \,dx
\to 0
\]
as $k\to +\infty$.
We recall the following well-known inequalities, that hold for any two 
real vectors $\xi, \eta$ and a real $p>1$:
\begin{equation}\label{ja}
 (|\xi|^{p-2}\xi-|\eta|^{p-2}\eta)(\xi-\eta)
\geq\begin{cases}
 2^{2-p}|\xi-\eta|^{p}, & \text{if } p\geq2, \\
 (p-1)\frac{|\xi-\eta|^{2}}{(|\xi|+|\eta|)^{2-p}}, & \text{if } 1<p<2.
 \end{cases}
\end{equation}
Therefore, 
\begin{equation} \label{ya}
\begin{aligned}
& 2^{2-p_i^{+}}\int_{\{x\in\Omega, p_i(x)\geq2\}} | D_i(u_{n_k}- u_n)|^{p_i(x)}
 \,dx \\
&\leq\int_{\{x\in\Omega, p_i(x)\geq2\}} \big[| D_i u_{n_k}|^{p_i(x)-2}
 D_i u_{n_k}-| D_i u_n|^{p_i(x)-2}D_i u_n\big] D_i(u_{n_k}- u_n) \,dx\\
&\leq\int_{\Omega} \big[| D_i u_{n_k}|^{p_i(x)-2}D_i u_{n_k}
 -| D_i u_n|^{p_i(x)-2}D_i u_n\big] D_i(u_{n_k}- u_n) \,dx\\
&=E_i(k).
\end{aligned}
\end{equation}
On the set $\Omega_i=\{x\in\Omega, 1<p_i(x)<2\}$, we employ \eqref{ja} as follows
\begin{equation}
\begin{aligned}
& \int_{\Omega_i}| D_iu_{n_k}- D_iu_n|^{p_i(x)} \,dx \\
&\leq\int_{\Omega_i} \frac{| D_iu_{n_k}- D_iu_n|^{p_i(x)}}
{\left(| D_iu_{n_k}|+| D_iu_n|\right)^{\frac{p_i(x)(2-p_i(x))}{2}}}
\left(| D_iu_{n_k}|+| D_iu_n|\right)^{\frac{p_i(x)(2-p_i(x))}{2}}\,dx\\
 &\leq 2\|\frac{| D_iu_{n_k}- D_iu_n|^{p_i(x)}}{\left(| D_iu_{n_k}|
+| D_iu_n|\right)^{\frac{p_i(x)(2-p_i(x))}{2}}}\|_{L^{\frac{2}{p_i(\cdot)}}
 (\Omega_i)} \\
&\quad \times\|\left(| D_iu_{n_k}|+| D_iu_n|\right)^{\frac{p_i(x)(2-p_i(x))}{2}}
\|_{L^{\frac{2}{2-p_i(\cdot)}}(\Omega)}\\
&\leq 2 \max\Big\{\Big( \int_{\Omega_i} \frac{| D_iu_{n_k}- D_iu_n|^{p_i(x)}}
{\big(| D_iu_{n_k}|+| D_iu_n|\big)^{\frac{p_i(x)(2-p_i(x))}{2}}}\,dx
 \Big)^{\frac{p_i^{-}}{2}}, \\
&\quad \Big( \int_{\Omega_i} \frac{| D_iu_{n_k}- D_iu_n|^{p_i(x)}}
{\left(| D_iu_{n_k}|+| D_iu_n|\right)^{\frac{p_i(x)(2-p_i(x))}{2}}}\,dx
 \Big)^{\frac{p_i^{+}}{2}}\Big\}\\
&\quad\times \max\Big\{\Big(\int_{\Omega} \big(| D_iu_{n_k}|+| D_iu_n|
 \big)^{p_i(x)}\,dx\Big)^{\frac{2-p_i^{+}}{2}}, \\
&\quad \Big(\int_{\Omega} \big(| D_iu_{n_k}|+| D_iu_n|\big)^{p_i(x)}\,dx
 \Big)^{\frac{2-p_i^{-}}{2}}\Big\}\\
&\leq 2 \max\Big\{(p_i^{-}-1)^{-\frac{p_i^{+}}{2}}E_i(k)^{\frac{p_i^{+}}{2}},
 (p_i^{-}-1)^{-\frac{p_i^{-}}{2}}E_i(k)^{\frac{p_i^{-}}{2}}\Big\}\\
&\quad\times \max\Big\{\Big(\int_{\Omega} \big(| D_iu_{n_k}|
+| D_iu_n|\big)^{p_i(x)}\,dx\Big)^{\frac{2-p_i^{+}}{2}}, \\
&\quad \Big(\int_{\Omega} \big(| D_iu_{n_k}|+| D_iu_n|\big)^{p_i(x)}\,dx
 \Big)^{\frac{2-p_i^{-}}{2}}\Big\}.
\end{aligned} \label{ba}
\end{equation}
Since $u_{n_k} $ is bounded in $\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ and
$u_n\in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$, after letting
$k\to+\infty$ in \eqref{ya} and \eqref{ba}, we find
\begin{equation*}
 \lim_{k\to+\infty}\int_{\Omega} | D_iu_{n_k}- D_iu_n|^{p_i(x)} \,dx=0,
\end{equation*}
which implies, for all $i=1,\dots ,N$,
\begin{equation}\label{e100}
D_iu_{n_k}\to D_iu_n \quad \text{strongly in }
 L^{p_i(\cdot)}(\Omega) \text{ and a.e. in } \Omega.
\end{equation}
We are going to prove \eqref{app-variationel} by passing to the limit in \eqref{12}.
By \eqref{e100} we have
$$
| D_i u_{n_k}|^{p_i(x)-2}D_iu_{n_k}\rightharpoonup
| D_i u_n|^{p_i(x)-2}D_iu_n \quad\text{weakly in }
 L^{p_i'(\cdot)}(\Omega),\; p_i'(\cdot)=\frac{p_i(\cdot)}{p_i(\cdot)-1}.
$$
From \eqref{e1} and Lebesgue's dominated convergence theorem, we obtain
$$
a_i(x,T_n(u_{n_k}))D_i\varphi\to a_i(x,T_n(u_n))D_i\varphi \quad
 \text{strongly in } L^{p_i(\cdot)}(\Omega), \; 1\leq i\leq N.
$$
Let $\rho_{j}(t)$ be an increasing, uniformly bounded Lipschitz function \cite{boc} 
(or $W ^{1,\infty}(\Omega)$ function), such that 
$\rho_{j}(\sigma) \to \chi_{\{|\sigma|>t\}}\operatorname{sign}(\sigma)$, as
 $j\to +\infty$. Taking $\rho_{j}(u_{n_k})$ as a test function in \eqref{12},
we obtain
\begin{align*}
& \sum^{N}_{i=1}\int_{\Omega}\rho'_{j}(u_{n_k}) a_i(x,T_n(u_{n_k}))
| D_iu_{n_k}|^{p_i(x)}\,dx+\int_{\Omega}T_k(| u_{n_k}|^{s(x)-1}
u_{n_k})\rho_{j}(u_{n_k}) \,dx\\
&=\int_{\Omega}T_n(f) \rho_{j}(u_{n_k}) \,dx.
\end{align*}
As $j\to+\infty$, we obtain
\begin{equation}\label{2-2}
\int_{\{| u_{n_k}|>t\}} | T_k(| u_{n_k}|^{s(x)-1}u_{n_k})| \,dx
\leq \int_{\{|u_{n_k}|>t\}}| f| \,dx.
\end{equation}
Let $E \subset \Omega$ be any measurable set, using \eqref{2-2}, we have
\begin{align*}
& \int_{E}| T_k(| u_{n_k}|^{s(x)-1}u_{n_k})| \,dx \\
&=\int_{E\cap\{|u_{n_k}|\leq t\}}| T_k(| u_{n_k}|^{s(x)-1}u_{n_k})| \,dx
 +\int_{E\cap\{| u_{n_k}|>t\}}| T_k(| u_{n_k}|^{s(x)-1}u_{n_k})| \,dx\\
&\leq (t^{s^{+}}+t^{s^{-}})\operatorname{meas}(E)
+\int_{\{| u_{n_k}|>t\}}| f| \,dx.
\end{align*}
Then we deduce that the sequence 
$\left\{T_k(| u_{n_k}|^{s(x)-1}u_{n_k})\right\}$ is equi-integrable in
$L^{1}(\Omega)$, and since 
$T_k(| u_{n_k}|^{s(x)-1}u_{n_k})\to|u_n|^{s(x)-1}u_n$
a.e. in $\Omega$, Vitali's theorem implies that
$$
T_k(| u_{n_k}|^{s(x)-1}u_{n_k})\to | u_n|^{s(x)-1}u_n\quad \text{in } L^{1}(\Omega).
$$
Therefore, we can obtain \eqref{app-variationel} by passing to the limit in 
\eqref{12}.

To show \eqref{est1}, we choose $\varphi=\frac{T_k(u_n)}{k}$ in 
\eqref{app-variationel} as a test function, we have
 $$ 
\int_{\Omega}|u_n|^{s(x)-1}u_n\frac{T_k(u_n)}{k}\,dx
\leq \int_{\Omega}T_n(f)\frac{T_k(u_n)}{k}\,dx\leq\int_{\Omega}|f|\,dx.
$$
 Fatou's lemma implies that estimate \eqref{est1} holds as $k\to0$.
\end{proof}

 In the rest of this paper, we will denote by $C_i$ (or $C$) the positive 
constants depending only on the data of the problem, but not on $n$.

\section{Uniform estimates}

In this section, we assume that $u_n$ is a solution of \eqref{approximate-equation}.

\begin{lemma}\label{lemma-2}
 Let $p_i:\overline{\Omega}\to(1,\infty)$, $s:\overline{\Omega}\to(0,\infty)$ 
and $\gamma_i:\overline{\Omega}\to[0,\infty)$ be continuous functions. 
Then, there exists a constant $C>0$ such that
\begin{gather}\label{estimation-11}
 \sum_{i=1}^{N}\int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}
{(1+| u_n |)^{\gamma_i(x)+\lambda}}\,dx\leq C,\quad \forall \lambda >1, \\
\label{tranestimate}
\int_\Omega|D_iT_k(u_n)|^{p_i(x)}\,dx\leq \frac{k}{\alpha}(1+k)^{\gamma_+^+}
\|f\|_{L^1(\Omega)},\quad i=1,\ldots,N.
\end{gather}
\end{lemma}

\begin{proof}
 We introduce the function $\psi:\mathbb{R}\to \mathbb{R}$ by
$$
\psi_{\lambda}(t)=\int_0^{t}\frac{dx}{(1+|x|)^{\lambda}}
=\frac{1}{1-\lambda}[(1+|t|)^{1-\lambda}-1]\operatorname{sign}(t),\ \lambda >1.
$$
Note that $\psi_{\lambda}$ is a continuous function satisfies 
$\psi_{\lambda}(0)=0$ and $|\psi'_{\lambda}(\cdot)|\leq 1$. 
We take $\psi_{\lambda}(u_n)$ as a test function in \eqref{app-variationel} 
and we use the assumption \eqref{a1-condition}, we obtain
\begin{equation*}
\sum_{i=1}^{N}\int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |
)^{\gamma_i(x)+\lambda}}\,dx\leq C_1\int_{\Omega}| f| \,dx,
\end{equation*}
In particular, there exists $C_2>0$ such that
\begin{equation}\label{estimation-111}
\int_{\Omega}\frac{| D_iu_n|^{p^{-}_i}}{(1+| u_n |)^{\gamma_{+}^{+}+\lambda}}\,dx
\leq C_2 , \quad \forall i=1,\dots ,N.
\end{equation}
We take $\varphi=T_k(u_n)$ in \eqref{app-variationel}, we find
$$
\int_\Omega\frac{|D_iT_k(u_n)|^{p_i(x)}}{(1+|u_n|)^{\gamma_i(x)}}\,dx
\leq \frac{k}{\alpha}\|f\|_{L^1(\Omega)}.
$$
Hence
\begin{align*}
\int_\Omega|D_iT_k(u_n)|^{p_i(x)}\,dx
&= \int_\Omega\frac{|D_iT_k(u_n)|^{p_i(x)}}{(1+|T_k(u_n)|)^{\gamma_i(x)}}
(1+|T_k(u_n)|)^{\gamma_i(x)}\,dx\\
&\leq \frac{k}{\alpha}(1+k)^{\gamma_+^+}\|f\|_{L^1(\Omega)}.
\end{align*}
Which yields \eqref{tranestimate}.
\end{proof}


\begin{lemma}\label{lemma-3} 
Assume that $s(\cdot)$, $p_i(\cdot)$ and $\gamma_i(\cdot)$ are restricted 
as in Theorem \ref{thm1}. Then, there exists a constant $C>0$
such that for all continuous functions $q_i(\cdot)$, $i=1,\dots ,N$ on 
$\overline{\Omega}$ as in \eqref{regularite-th2}, we have
\begin{gather}\label{est3}
 \| D_iu_n\|_{L^{q_i(\cdot)}(\Omega)}\leq C, \\
\label{est33}
 \| u_n\|_{L^{\overline{q}^{\star}(\cdot)}(\Omega)}\leq C.
\end{gather}
\end{lemma}

\begin{proof}
 Firstly, for $p_i$ is defined in \eqref{cond-th2-p}, we have
\begin{equation*}
 1<\frac{N(\overline{p}(x)-1-\gamma^{+}_{+})p_i(x)}
{\overline{p}(x)(N-1-\gamma^{+}_{+})},\quad \forall x\in \overline{\Omega}.
\end{equation*}
By \eqref{regularite-th2} and \eqref{n-condition}, we deduce
$ q_i(x)<p_i(x)$ for all $x\in \overline{\Omega}$, $i=1,\dots ,N$.
\smallskip

\noindent\textbf{Case (a):}
 In the first step, let $q_i^{+}$ be a constant satisfying
\begin{equation}\label{q+}
 q_i^{+}<\frac{N(\overline{p}^{-}-1-\gamma^{+}_{+})p_i^{-}}{\overline{p}^{-}
(N-1-\gamma^{+}_{+})},\quad  i=1,\dots ,N,\; 
\frac{1}{\overline{p}^{-}}=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{p_i^{-}}.
\end{equation}
We can assume that $\frac{q_i^{+}}{p_i^{-}}=\frac{\overline{q}^{+}}{\overline{p}^{-}}$,
where $\frac{1}{\overline{q}^{+}}=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{q_i^{+}}$. 
If not, we set $\theta=\max\{q_i^{+}/p_i^{-}$, $i=1,\dots ,N\}$ and replace 
$q_i^{+}$ by $\theta p_i^{-}$. Observe that, since
 $\theta p_i^{-}\geq q_i^{+}$, the fact that $(D_iu_n$) remains in a bounded set 
of $L^{\theta p_i^{-}}(\Omega)$ implies the result.

From now on, we set $q_i^{+}=\theta p_i^{-}$,
$\theta=\frac{\overline{q}^{+}}{\overline{p}^{-}}\in 
(0,\frac{N(\overline{p}^{-}-1-\gamma^{+}_{+})}{\overline{p}^{-}
(N-1-\gamma^{+}_{+})})\subseteq(0,1)$. Then \eqref{q+} is equivalent to
$$
 \big(\frac{1-\theta}{\theta}\big)\overline{q}^{+\star}-\gamma^{+}_{+}>1,\quad 
\overline{q}^{+\star}=\frac{N\overline{q}^{+}}{N-\overline{q}^{+}}.
$$
Hence there exists $\lambda>1$ such that
\begin{equation*}
 \big(\frac{1-\theta}{\theta}\big)\overline{q}^{+\star}-\gamma^{+}_{+}>\lambda>1,
\end{equation*}
so,
\begin{equation}\label{q*}
 (\gamma^{+}_{+}+\lambda)\big(\frac{\theta}{1-\theta}\big)<\overline{q}^{+\star}.
\end{equation}
Using H\"{o}lder's inequality and \eqref{estimation-111}, we obtain
\begin{align*}
\int_{\Omega}| D_iu_n|^{q^{+}_i}\,dx
&=\int_{\Omega}\frac{| D_iu_n|^{q^{+}_i}}{(1+| u_n |)^{(\gamma_{+}^{+}
 +\lambda)\theta}}(1+| u_n |)^{(\gamma_{+}^{+}+\lambda)\theta} \,dx \\
&\leq \Big(\int_{\Omega}\frac{| D_iu_n|^{p^{-}_i}}{(1+| u_n |)^{\gamma_{+}^{+}
 +\lambda}}\,dx\Big)^{\theta}
\Big(\int_{\Omega}(1+| u_n |)^{(\gamma_{+}^{+}+\lambda)\frac{\theta}{1-\theta}}
 \Big)^{1-\theta} \\
&\leq C \Big(\int_{\Omega}(1+| u_n |)^{(\gamma_{+}^{+}+\lambda)
 \frac{\theta}{1-\theta}}\Big)^{1-\theta},
\end{align*}
so that
$$
\prod_{i=1}^{N}\Big(\| D_iu_n\|_{L^{q_i^{+}}(\Omega)}\Big)^{1/N}
\leq C^{\frac{1}{\overline{q}^+}} 
\Big(\int_{\Omega}(1+| u_n |)^{(\gamma_{+}^{+}+\lambda)\frac{\theta}{1-\theta}}
\Big)^{\frac{1-\theta}{\overline{q}^+}}.
$$
Therefore, by \eqref{q*} and Young's inequality, we can write
\begin{equation}\label{estimation}
\prod_{i=1}^{N}\Big(\| D_iu_n\|_{L^{q_i^{+}}(\Omega)}\Big)^{1/N}
\leq C_1(\varepsilon)+\varepsilon
\Big(\int_{\Omega}| u_n |^{\overline{q}^{+\star}}\Big)
^{\frac{1-\theta}{\overline{q}^+}}.
\end{equation}
In view of \eqref{anisotropic-inequality-1}, with $r = \overline{q}^{+\star}$,
 we obtain
\begin{equation}\label{eq}
 \| u_n\|_{L^{\overline{q}^{+\star}}(\Omega)}
\leq C_0\prod_{i=1}^{N}\Big(\| D_iu_n\|_{L^{q_i^{+}}(\Omega)}\Big)^{1/N}
\leq C_2(\varepsilon)+\varepsilon C_0
\big(\| u_n\|_{L^{\overline{q}^{+\star}}(\Omega)}\big)
^{\frac{ N(1-\theta)}{N-\overline{q}^{+}}}.
\end{equation}
We choose $\varepsilon=1/(2C_0)$, then
\begin{equation}\label{anisotropic2}
 \| u_n\|_{L^{\overline{q}^{+\star}}(\Omega)}
\leq C_{3}+\frac{1}{2}\| u_n\|_{L^{\overline{q}^{+\star}}(\Omega)}^{\eta},\quad
 \eta=(1-\theta)\frac{N}{N-\overline{q}^{+}}.
\end{equation}
The assumption \eqref{n-condition} implies that $\eta\in(0,1]$. 
Hence, the estimate \eqref{anisotropic2} implies \eqref{est33}, and by
 \eqref{estimation} we deduce that \eqref{est3} holds. 
This completes the proof of the case (a).
\smallskip

\noindent\textbf{Case (b):} In the second, we suppose that \eqref{regularite-th2} 
holds and
\begin{equation*}
 q_i^{+}\geq\frac{N(\overline{p}^{-}-1-\gamma^{+}_{+})p_i^{-}}{\overline{p}^{-}
(N-1-\gamma^{+}_{+})}.
\end{equation*}
 By the continuity of $p_i(\cdot)$ and $q_i(\cdot)$ on $\overline{\Omega}$, 
there exists a constant $\delta>0$ such that
\begin{equation}\label{min}
 \max_{t\in\overline{ Q(x,\delta)\cap\Omega}} q_i(t)
<\min_{t\in\overline{ Q(x,\delta)\cap\Omega}} 
\frac{N(\overline{p}(t)-1-\gamma^{+}_{+})p_i(t)}{\overline{p}(t)
(N-1-\gamma^{+}_{+})},\quad\forall\ x\in\Omega,
\end{equation}
where $Q(x,\delta)$ is a cube with center $x$ and diameter $\delta$. 
Note that $\overline{\Omega}$ is compact and therefore we can cover it 
with a finite number of cubes $(Q_{j})_{j=1,\dots ,k}$ with edges parallel 
to the coordinate axes. Moreover there exists a constant $\nu>0$ such that
\begin{equation*}
 \delta>|\Omega_{j}|=\operatorname{meas}(\Omega_{j})>\nu,\quad
 \Omega_{j}=Q_{j}\cap\Omega\quad \text{for all } j=1,\dots ,k.
\end{equation*}
We denote by $q_{i,j}^{+}$ the local maximum of $q_i(\cdot)$ on 
$\overline{\Omega_{j}}$ (respectively $p_{i,j}^{-}$ the local minimum of 
$p_i(\cdot)$ on $\overline{\Omega_{j}}$), such that
\begin{equation}\label{local}
 q_{i,j}^{+}<\frac{N(\overline{p}_{j}^{-}-1-\gamma^{+}_{+})p_{i,j}^{-}}
{\overline{p}_{j}^{-}(N-1-\gamma^{+}_{+})}\quad \text{for all }
j=1,\dots ,k,\; \frac{1}{\overline{p}_{j}^{-}}=\frac{1}{N}
\sum_{i=1}^{N}\frac{1}{p_{i,j}^{-}}.
\end{equation}
By \eqref{anisotropic-inequality-2}, we have
\begin{equation}\label{estimo}
 \| u_n\|_{L^{\overline{q}_{j}^{+\star}}(\Omega_{j})}
\leq C_1\prod_{i=1}^{N}\Big(\| u_n\|_{L^{q_{i,j}^{+}}
(\Omega_{j})}+\| D_iu_n\|_{L^{q_{i,j}^{+}}(\Omega_{j})}\Big)^{1/N}.
\end{equation}
We combine \eqref{cond-th2-s}, \eqref{est1}, \eqref{estimo}, and the fact
 that $q_{i,j}^{+}<p_{i,j}^{-}\leq s_{j}^{-}=\min_{x\in\overline{\Omega_{j}}}s(x)$, 
we obtain
\begin{equation}\label{estima}
 \| u_n\|_{L^{\overline{q}_{j}^{+\star}}(\Omega_{j})}
\leq C_2\prod_{i=1}^{N}\Big(1+\| D_iu_n\|_{L^{q_{i,j}^{+}}(\Omega_{j})}\Big)^{1/N}.
\end{equation}
Now, arguing locally as in \eqref{estimation} and \eqref{eq}, we obtain
\begin{equation}\label{net}
 \| u_n\|_{L^{\overline{q}_{j}^{+\star}}(\Omega_{j})}
\leq C_2\prod_{i=1}^{N}\Big(1+\| D_iu_n\|_{L^{q_{i,j}^{+}}(\Omega_{j})}\Big)^{1/N}
\leq C_{3}+\frac{1}{2}\| u_n\|^{\eta_{j}}_{L^{\overline{q}_{j}^{+\star}}(\Omega_{j})},
\end{equation}
where
\begin{equation*}
\eta_{j}=\Big(1-\frac{\overline{q}_{j}^{+}}{\overline{p}_{j}^{-}}\Big) 
\frac{N}{N-\overline{q}_{j}^{+}}.
\end{equation*}
Thanks to \eqref{n-condition}, we have $\eta_{j}\in(0,1]$. 
Hence, the estimate \eqref{net} implies 
\begin{gather}
\int_{\Omega_{j}}| u_n|^{\overline{q}^{+\star}_{j}}\,dx\leq C_4\quad
 \text{for  } j=1,\dots ,k, \nonumber \\
\label{estimu}
\int_{\Omega_{j}}| D_iu_n|^{q^{+}_{i,j}}\,dx\leq C_5\quad \text{for } j=1,\dots ,k.
\end{gather}
Knowing that $q_i(x)\leq q^{+}_{i,j}$ and 
$\overline{q}^{\star}(x)\leq \overline{q}^{+\star}_{j}$ for all 
$x\in\overline{\Omega}_{j}$ and for $j=1,\dots ,k$, we conclude that
\begin{equation*}
\int_{\Omega_{j}}| u_n|^{\overline{q}^{\star}(x)}\,dx
+\int_{\Omega_{j}}| D_iu_n|^{q_i(x)}\,dx\leq C_6,
\end{equation*}
which finally implies
\begin{equation*}
 \int_{\Omega}| u_n|^{\overline{q}^{\star}(x)}\,dx
+\int_{\Omega}| D_iu_n|^{q_i(x)}\,dx
\leq\sum_{j=1}^{k}\Big(\int_{\Omega_{j}}| u_n|^{\overline{q}^{\star}(x)}\,dx
+\int_{\Omega_{j}}| D_iu_n|^{q_i(x)}\,dx\Big)\leq C.
\end{equation*}
Where $C$ is a constant independent of $n$. 
This finishes the proof of lemma \ref{lemma-3}.
\end{proof}

\begin{lemma}\label{lemma-4} 
Let $m , s , p_i$ and $\gamma_i$ be restricted as in Theorem \ref{thm4}.
Then, there exists a constant $C>0$ such that
\begin{equation}\label{est2}
 \sum_{i=1}^{N}\int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)
+1-[(m(x)-1)s(x)]}}\,dx+\int_{\Omega}|u_n|^{m(x)s(x)}\,dx\leq C.
\end{equation}
\end{lemma}

\begin{proof} 
Taking $\psi(x,u_n)=\left((1+| u_n |)^{(m(x)-1)s(x)}-1\right)\operatorname{sign}(u_n)$
in \eqref{approximate-equation} as a test function, by \eqref{a1-condition} and 
the fact that for a.e. $x\in\Omega$ and for all $i=1,\dots ,N$
\begin{align*}
 D_i\psi(x,u_n)
&= (m(x)-1)(1+| u_n |)^{(m(x)-1)s(x)}\operatorname{sign}(u_n)D_is(x)\ln(1+| u_n |)\\
&\quad +\frac{(m(x)-1)s(x)D_iu_n}{(1+| u_n |)^{1-(m(x)-1)s(x)}}\\
&\quad +D_i m(x)(1+| u_n |)^{(m(x)-1)s(x)}\operatorname{sign}(u_n) s(x)\ln(1+| u_n |),
\end{align*}
we obtain
\begin{align*}
& \alpha s^{-}(m^{-}-1)\sum_{i=1}^{N}
\int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)+1-((m(x)-1)s(x))}}\,dx\\
 & +\int_{\Omega}|u_n|^{s(x)}\left((1+| u_n |)^{(m(x)-1)s(x)}-1\right)\,dx\\
 &\leq \int_{\Omega}|f|\left((1+| u_n |)^{(m(x)-1)s(x)}-1\right)\,dx\\
 &\quad+C_1\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{(m(x)-1)s(x)}
 \ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx.
\end{align*}
Using that $| u_n |^{s(x)}\geq\min\{1; 2^{1-s^{+}}\}(1+| u_n |)^{s(x)}-1$,
 Proposition \ref{pro1}, and Young inequality, we have
\begin{equation} \label{es}
\begin{aligned}
& \sum_{i=1}^{N}\int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |
 )^{\gamma_i(x)+1-((m(x)-1)s(x))}}\,dx
+\frac{1}{2}\int_{\Omega}(1+| u_n |)^{m(x)s(x)}\,dx\\
&\leq C_2+C_{3}\max\big(\|f\|_{L^{m(\cdot)}(\Omega)}^{m^{+}},
 \|f\|_{L^{m(\cdot)}(\Omega)}^{m^{-}}\big)\\
&\quad +C_4\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{(m(x)-1)s(x)}
 \ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx.
\end{aligned}
\end{equation}
We can estimate the last term in \eqref{es} by applying  Young's inequality
\begin{equation} \label{lb}
\begin{aligned}
& \int_{\Omega}(1+| u_n |)^{(m(x)-1)s(x)}\ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx\\
&=\int_{\Omega}(1+| u_n |)^{\frac{(m(x)-1)s(x)+(p_i(x)-1)(\gamma_i(x)+1)}{p_i(x)}}
\ln(1+| u_n |) \\
&\quad\times \frac{| D_iu_n|^{p_i(x)-1}}{(1+| u_n |
 )^{\frac{\gamma_i(x)+1-(m(x)-1)s(x)}{p'_i(x)}}}\,dx\\
& \leq C_5\int_{\Omega}(1+|u_n|)^{(m(x)-1)s(x)
 +(p_i(x)-1)(\gamma_i(x)+1)}\left(\ln(1+| u_n |)\right)^{p_i(x)}\,dx\\
&\quad + \frac{1}{4C_4}\int_{\Omega}
 \frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)+1-(m(x)-1)s(x)}}\,dx.
\end{aligned}
\end{equation}
We combine \eqref{es} and \eqref{lb}, we obtain
\begin{equation}\label{gig}
\begin{aligned}
& \frac{3}{4}\sum_{i=1}^{N}\int_{\Omega}
 \frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)+1-(m(x)-1)s(x)}}\,dx
 +\frac{1}{2}\int_{\Omega}(1+| u_n |)^{m(x)s(x)}\,dx\\
&\leq C_6+C_7\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{(m(x)-1)s(x)+(p_i(x)-1)
 (\gamma_i(x)+1)} \\
&\quad\times \ln(1+ |u_n|)^{p_i(x)}\,dx 
=I.
\end{aligned}
\end{equation}
Since $s(x)>(p_i(x)-1)(\gamma_i(x)+1)$, we have
$$
(p_i(x)-1)(\gamma_i(x)+1)-s(x)\leq((p_i(x)-1)(\gamma_i(x)+1)-s(x))^{+}
=b_i<\frac{b_i}{2}<0,
$$
and $(1+|t|)^{(p_i(x)-1)(\gamma_i(x)+1)-s(x)-\frac{b_i}{2}}\ln(1+ |t|)^{p_i(x)}$ 
is bounded for all $x\in\overline{\Omega}$ and $t\in\mathbb{R}$.
We conclude that
\begin{align*}
& (1+| u_n |)^{(m(x)-1)s(x)+(p_i(x)-1)(\gamma_i(x)+1)}
 \ln(1+ |u_n|)^{p_i(x)} =(1+| u_n |)^{m(x)s(x)+\frac{b_i}{2}} \\
& \times (1+| u_n |)^{(p_i(x)-1)(\gamma_i(x)+1)-s(x)-\frac{b_i}{2}}
 \ln(1+ |u_n|)^{p_i(x)}\\
&\leq C(1+| u_n |)^{m(x)s(x)+\frac{b_i}{2}}.
\end{align*}
By another application of Young inequality, we obtain
\begin{equation}\label{obt1}
 I\leq \frac{1}{8}\int_{\Omega}(1+| u_n |)^{m(x)s(x)}\,dx+C_{8}.
\end{equation}
Therefore, by \eqref{gig} and \eqref{obt1} we obtain the estimation \eqref{est2}.
\end{proof}

\begin{lemma}\label{lemma-5} 
Let $p_i$, $s$ and $\gamma_i$ be restricted as in Theorem \ref{thm2}.
Then, there exists a positive constant $C$ such that
\begin{equation}\label{est-lemma-5}
 \|D_iu_n\|_{L^{q_i(\cdot)}(\Omega)}\leq C,
\end{equation}
for all continuous functions $q_i$ on $\overline{\Omega}$ satisfying 
\eqref{regularite-1}.
\end{lemma}

\begin{proof}
 Note that, assumption \eqref{regularite-1} implies that 
$q_i(x)<p_i(x)$ for all $x\in \overline{\Omega}$, $i=1,\dots ,N$. We can write
\begin{equation*}
 \int_{\Omega}| D_iu_n|^{q_i(x)}\,dx
=\int_{\Omega}\frac{| D_iu_n|^{q_i(x)}}{(1+| u_n |
)^{\frac{q_i(x)}{p_i(x)}(\gamma_i(x)+\lambda)}}(1+| u_n |
)^{\frac{q_i(x)}{p_i(x)}(\gamma_i(x)+\lambda)}\,dx.
\end{equation*}
 Using Young inequality and \eqref{estimation-11}, we obtain
\begin{equation}\label{ss}
 \int_{\Omega}| D_iu_n|^{q_i(x)}\,dx
\leq C_1+C_2\int_{\Omega}(1+| u_n |
)^{\frac{q_i(x)(\gamma_i(x)+\lambda)}{p_i(x)-q_i(x)}}\,dx.
\end{equation}
Then assumption \eqref{regularite-1} implies 
$\frac{s(x)(p_i(x)-q_i(x))}{q_i(x)}-\gamma_i(x)>1$. Choosing
\begin{equation*}
 \lambda=\min_{1\leq i\leq N}\min_{x\in\overline{\Omega}}
\Big(\frac{s(x)(p_i(x)-q_i(x))}{q_i(x)}-\gamma_i(x)\Big)>1\,.
\end{equation*}
Thanks to the choice of $\lambda$ and \eqref{regularite-1}, we have
\begin{equation}\label{sm}
 \frac{q_i(x)(\gamma_i(x)+\lambda)}{p_i(x)-q_i(x)}\leq s(x),\quad
 \forall x\in\overline{\Omega},\; \forall i=1,\dots ,N.
\end{equation}
Combining \eqref{sm}, \eqref{ss}, and \eqref{est1} results 
\eqref{est-lemma-5}.
\end{proof}

\begin{lemma}\label{lemma-6} 
Let $m$, $s$, $p_i$, and $\gamma_i$ be restricted as in Theorem \ref{thm4}. 
Then, there exists a constant $C>0$ such that
\begin{equation}\label{est-lemma-6}
 \|D_iu_n\|_{L^{q_i(\cdot)}(\Omega)}\leq C,
\end{equation}
for all continuous functions $q_i$ on $\overline{\Omega}$ satisfying 
\eqref{regularite-th-4}.
\end{lemma}

\begin{proof} 
Note that $s(x)<\frac{1+\gamma_{+}(x)}{m(x)-1}$ and \eqref{regularite-th-4}
 imply $q_i(x)<p_i(x)$. Then by Young's inequality, we have
 \begin{align*}
& \int_{\Omega}| D_iu_n|^{q_i(x)}\,dx\\
&=\int_{\Omega}\frac{| D_iu_n|^{q_i(x)}}{(1+| u_n |
 )^{(\gamma_i(x)+1-[(m(x)-1)s(x)])}
 \frac{q_i(x)}{p_i(x)}}(1+| u_n |)^{(\gamma_i(x)+1-[(m(x)-1)s(x)])
 \frac{q_i(x)}{p_i(x)}}\,dx \\
&\leq \int_{\Omega}\Big(\frac{q_i(x)}{p_i(x)}\Big)
\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)+1-[(m(x)-1)s(x)]}}\,dx\\
&\quad +\int_{\Omega}\Big(1-\frac{q_i(x)}{p_i(x)}\Big)
 (1+|u_n|)^{\frac{(\gamma_i(x)+1-[(m(x)-1)s(x)])q_i(x)}{p_i(x)-q_i(x)}}\,dx,
 \end{align*}
and by \eqref{regularite-th-4}, we obtain
\begin{equation}\label{ls}
\begin{aligned}
&\int_{\Omega}| D_iu_n|^{q_i(x)} \,dx \\
&\leq C_1 \int_{\Omega}\frac{| D_iu_n|^{p_i(x)}}{(1+| u_n |)^{\gamma_i(x)+1
-[(m(x)-1)s(x)]}}\,dx
+C_2\int_{\Omega}(1+|u_n|)^{m(x)s(x)}\,dx.
\end{aligned}
\end{equation}
Therefore, \eqref{ls} and \eqref{est2} imply the desired result.
\end{proof}

\begin{lemma}\label{lemma-7}
 Let $m$, $s$, $p_i$, and $\gamma_i$ be restricted as in Theorem \ref{thm3}.
Then, there exists a constant $C>0$ such that
\begin{equation}\label{est-lemma-7}
 \sum_{i=1}^{N}\int_{\Omega}| D_iu_n|^{p_i(x)} \,dx
+\int_{\Omega}| u_n |^{s(x)+1+\gamma_{+}(x)}\,dx\leq C.
\end{equation}
\end{lemma}

\begin{proof} 
Taking $\psi(x,u_n)=\left((1+| u_n |)^{1+\gamma_{+}(x)}-1\right)
\operatorname{sign}(u_n)$ 
in \eqref{approximate-equation} as a test function, by \eqref{a1-condition} and 
the fact that for a.e. $x\in\Omega$, for all $i=1,\dots , N$,
\begin{align*}
 D_i\psi(x,u_n)
&= (1+\gamma_{+}(x))(1+| u_n |)^{\gamma_{+}(x)}D_iu_n \\
&\quad +D_i\gamma_{+}(x)(1+| u_n |)^{1+\gamma_{+}(x)} \ln(1+| u_n |)\operatorname{sign}(u_n),
\end{align*}
we set $\gamma_{+}^{-}=\max_{1\leq i\leq N}\min_{x\in\overline{\Omega}}\gamma_i(x)$, 
we obtain
\begin{align*}
&\alpha (1+\gamma_{+}^{-})\sum_{i=1}^{N}\int_{\Omega}| D_iu_n|^{p_i(x)}\,dx
 +\int_{\Omega}|u_n|^{s(x)}\left((1+| u_n |)^{1+\gamma_{+}(x)}-1\right)\,dx\\
&\leq \int_{\Omega}|f|\left((1+| u_n |)^{1+\gamma_{+}(x)}-1\right)\,dx\\
&\quad +C_1\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{1+\gamma_{+}(x)}
\ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx.
\end{align*}
Using  that $| u_n |^{s(x)}\geq \min \{1,2^{1-s^{+}}\}(1+| u_n |)^{s(x)}-1$, 
Proposition \ref{pro1} and Young's inequality and since 
$m'(\cdot)(1+\gamma_{+}(x))\leq s(x)+1+\gamma_{+}(x)$, we have
\begin{equation} \label{esm}
\begin{aligned}
& \sum_{i=1}^{N}\int_{\Omega}| D_iu_n|^{p_i(x)}
 +\frac{1}{2}\int_{\Omega}(1+| u_n |)^{s(x)+1+\gamma_{+}(x)}\,dx\\
&\leq C_2+C_{3}\max\left(\|f\|_{L^{m(\cdot)}(\Omega)}^{m^{+}},
 \|f\|_{L^{m(\cdot)}(\Omega)}^{m^{-}}\right)\\
&\quad +C_4\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{1+\gamma_{+}(x)}
\ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx.
\end{aligned}
\end{equation}
We can estimate the last term in \eqref{esm} by applying Young's inequality,
\begin{equation} \label{lbn}
\begin{aligned}
& \int_{\Omega}(1+| u_n |)^{1+\gamma_{+}(x)}\ln(1+| u_n |)| D_iu_n|^{p_i(x)-1}\,dx\\
&\leq C_5\int_{\Omega}(1+|u_n|)^{p_i(x)(1+\gamma_{+}(x))}
 \left(\ln(1+| u_n |)\right)^{p_i(x)}
  + \frac{1}{2C_4}\int_{\Omega}| D_iu_n|^{p_i(x)}\,dx.
\end{aligned}
\end{equation}
We combine \eqref{esm} and \eqref{lbn} to obtain
\begin{equation} \label{c3}
\begin{aligned}
& \sum_{i=1}^{N}\int_{\Omega}| D_iu_n|^{p_i(x)}\,dx
 +\int_{\Omega}(1+| u_n |)^{s(x)+1+\gamma_{+}(x)}\,dx\\
& \leq C_6+C_7\sum_{i=1}^{N}\int_{\Omega}(1+| u_n |)^{p_i(x)
(\gamma_i(x)+1)}\ln(1+ |u_n|)^{p_i(x)}\,dx=J.
\end{aligned}
\end{equation}
Thanks to Remark \ref{rem} we have $s(x)>(p_i(x)-1)(\gamma_{+}(x)+1)$, so
$$
(p_i(x)-1)(\gamma_{+}(x)+1)-s(x)\leq((p_i(x)-1)(\gamma_{+}(x)+1)-s(x))^{+}
=d_i<\frac{d_i}{2}<0,
$$
and $(1+|t|)^{(p_i(x)-1)(\gamma_{+}(x)+1)-s(x)-\frac{d_i}{2}}\ln(1+ |t|)^{p_i(x)}$
 is bounded for all $x\in\overline{\Omega}$ and $t\in\mathbb{R}$.
We write
\begin{align*}
& (1+| u_n |)^{p_i(x)(\gamma_{+}(x)+1)}\ln(1+ |u_n|)^{p_i(x)} \\
&=(1+| u_n |)^{s(x)+\gamma_{+}(x)+1+\frac{d_i}{2}} 
  (1+| u_n |)^{(\gamma_i(x)+1)(p_i(x)-1)-s(x)
 -\frac{d_i}{2}}\ln(1+ |u_n|)^{p_i(x)}.
\end{align*}
By another application of Young's inequality, we obtain
\begin{equation}\label{obt}
 J\leq \frac{1}{4}\int_{\Omega}(1+| u_n |)^{s(x)+1+\gamma_{+}(x)}\,dx+C_{8}.
\end{equation}
Using \eqref{c3} and \eqref{obt}, we obtain  \eqref{est-lemma-7}.
\end{proof}

\begin{lemma}\label{gradient} 
Let $f_n\in L^{\infty}(\Omega)$ be a sequence of functions which is
 strongly convergent to some $f$ in $L^{1}(\Omega)$ and let $u_n$ 
be a solution of the problem
\begin{equation}\label{p1}
\begin{gathered}
-\sum_{i=1}^{N}D_i\left(a_i(x,T_n(u_n))|D_iu_n|^{p_i(x)-2}D_iu_n\right)
=f_n\quad \text{in } \Omega,\\
u_n=0\quad  \text{on }  \partial\Omega.
\end{gathered}
\end{equation}
Suppose that:
\begin{itemize}
\item[(i)] $u_n$ is such that $T_k(u_n)\in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ 
 for all $k>0$.

\item[(ii)] $u_n$ converges almost everywhere in $\Omega$ to some measurable 
function $u$ which is finite almost everywhere, and such that 
$T_k(u)\in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ for all $k>0$ (note that (i) and 
(ii) imply that $T_k(u_n)$ weakly converges to $T_k(u)$ in 
$\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$).

\item[(iii)] $u_n$ is bounded in $M^{r_1} (\Omega)$ for some $r_1 > 0$ and 
$u\in M^{r_1} (\Omega)$.

\item[(iv)] There exists $\theta_i> 0$ , $i=1,\dots ,N$ such that 
$| D_iu_n|^{\theta_i}$ is bounded in $L^{r_2} (\Omega)$, for some $r_2 > 1$
 and $| D_iu|^{\theta_i}\in L^{r_2} (\Omega)$.
\end{itemize}
Then, up to a subsequence, $D_iu_n$ converges to $D_i u $ almost everywhere 
in $\Omega$ for all $i=1,\dots ,N$.
\end{lemma}

\begin{proof} 
It has been proved in \cite{fan1} that there exists a solution 
$u_n \in\mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$ to problem \eqref{p1}. 
We follow the technique in \cite{Al, xi} with some modifications, 
since our method depends on the anisotropic variable exponent.
Define the vector-valued function 
$\hat{a}(x,s,\xi):\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}^{N}$, where 
$\hat{a}(x,s,\xi)=\{ \hat{a}_i(x,s,\xi)\}_{i=1,\dots ,N}$ with 
$\hat{a}_i(x,s,\xi)=a_i(x,s)|\xi_i|^{p_i(x)-2}\xi_i$.
Let $\theta$ be a real number between $0$ and $1$, which will be chosen later, and
\begin{equation*}
 I(n)=\int_{\Omega}\big\{\big( \hat{a}(x,T_n(u_n),\nabla u_n)
-\hat{a}(x,T_n(u_n),\nabla u)\big)\nabla( u_n-u)\big\}^{\theta}\,dx.
\end{equation*}
Note that $I(n)$ is well defined and $I(n)\geq0$. We fixe $k > 0$ and 
split the integral $I(n)$ on the sets $\left\{ | u| \geq k\right\}$ and 
$\{ | u| < k\}$,
obtaining
\begin{gather*}
I_1(n,k)=\int_{\{ | u| \geq k\}}
\big\{\big(\hat{a}(x,T_n(u_n),\nabla u_n)-\hat{a}(x,T_n(u_n),\nabla u)\big)
\nabla(u_n-u) \big\}^{\theta}\,dx, \\
I_2(n,k)=\int_{\{ | u| < k\}}
\big\{\big( \hat{a}(x,T_n(u_n),\nabla u_n)-\hat{a}(x,T_n(u_n),\nabla u)\big)
\nabla( u_n-u)\big\}^{\theta}\,dx.
\end{gather*}
By condition \eqref{a1-condition} and Young's inequality, we have
\begin{align*}
 I_1(n,k)
&\leq C_1\int_{\left\{ | u| \geq k\right\}}
\Big\{\sum_{i=1}^{N}\left(| D_i u_n|^{p_i(x)}+ | D_i u|^{p_i(x)}\right)
 \Big\}^{\theta}\,dx,\\
&\leq C_1\sum_{i=1}^{N}\int_{\left\{ | u| \geq k\right\}}
\left(2+| D_i u_n|^{\theta p_i^{+}}+ | D_i u|^{\theta p_i^{+}}\right)\,dx.
\end{align*}
We now choose $\theta < 1$ such that $\theta p_i^{+} <\theta_i$, 
$i=1,\dots ,N$. Using the H\"older inequality and (iv), we
obtain
\begin{align*}
I_1(n,k)
&\leq C_2\sum_{i=1}^{N} \bigg(\Big(\int_{\Omega}| D_i u_n|^{\theta_ir_2}\,dx
\Big)^{\frac{1}{r_2}}+ \Big(\int_{\Omega}| D_i u|^{\theta_ir_2}\,dx
\Big)^{\frac{1}{r_2}}\bigg)|\left\{ | u| \geq k\right\}|^{1-\frac{1}{r_2}}\\
&\quad +C_2|\{ |u| \geq k\}|
 \leq C|\{ |u| \geq k\}|^{1-\frac{1}{r_2}}  +C_2|\left\{ | u| \geq k\right\}|.
\end{align*}
By (ii),  for any $k>1$, we have
$$
|\left\{ | u| \geq k\right\}|\leq |\left\{ | u| > k-1\right\}|
\leq \frac{C}{(k-1)^{r_1}}.
$$
Using the above inequality, we obtain
\begin{equation}\label{18}
\lim_{k\to +\infty}\limsup_{n\to +\infty}I_1(n,k)=0.
\end{equation}
As $\nabla u = \nabla T_k(u)$ on the set $\{ | u| < k\}$, we obtain
 $$
I_2(n,k)=\int_{\left\{ | u| < k\right\}}
\big\{\big(\hat{a}(x,T_n(u_n), \nabla u_n)-\hat{a}(x,T_n(u_n),\nabla T_k(u))\big)
\nabla( u_n-T_k(u)) \big\}^{\theta}\,dx.
$$
Take $h > k+1$ and split the integral $I_2(n,k)$ on the sets 
$\left\{ | u_n-T_k( u)| \geq h\right\}$ and 
$\left\{ | u_n-T_k( u)| < h\right\}$, obtaining
\begin{align*} 
&I_{3}(n,k,h) \\
&=\int_{\left\{ | u_n-T_k( u)| \geq h\right\}}
\big\{\big( \hat{a}(x,T_n(u_n),\nabla u_n)-\hat{a}(x,T_n(u_n),
\nabla T_k(u))\big) \nabla( u_n-T_k(u))\big\}^{\theta}\,dx,
\end{align*}
and
\begin{align*}
I_4(n,k,h)
&=\int_{\left\{ | u_n-T_k( u)| < h\right\}}
\Big\{ \Big(\hat{a}(x,T_n(u_n),\nabla u_n) \\
&\quad -\hat{a}(x,T_n(u_n),\nabla T_k(u))\Big)
\nabla( u_n-T_k(u))\Big\}^{\theta}\,dx.
\end{align*}
As $|u_n|\geq h-k$ on the set $\left\{ | u_n-T_k( u)| \geq h\right\}$, we obtain
$$
|\left\{ | u_n-T_k( u)| \geq h\right\}|
\leq |\left\{ | u_n| \geq h-k\right\}|\leq \frac{C}{(h-k-1)^{r_1}}.
$$
Similarly to the discussion of $I_1(n, k)$ (with the same choice of $\theta$),
 we obtain
\begin{equation}\label{19}
\lim_{h\to +\infty}\limsup_{k\to +\infty}\limsup_{n\to +\infty}I_{3}(n,k,h)=0.
\end{equation}
Since $\nabla (u_n-T_k(u))=\nabla T_{h}(u_n-T_k(u))$ on the set 
$\left\{ | u_n-T_k( u)| < h\right\}$,
by H\"older inequality (with exponents $\frac{1}{\theta}$ and 
$\frac{1}{1-\theta}$), we have
\begin{align*}
 I_4(n,k,h)
&\leq|\Omega|^{1-\theta}\Big\{
\int_{\{|u_n-T_k(u)|<h\}} \Big(\hat{a}(x,T_n(u_n),\nabla u_n)\\
&\quad -\hat{a}(x,T_n(u_n),\nabla T_k(u))\Big)
\nabla T_{h}( u_n-T_k(u))\,dx\Big\}^{\theta}.
\end{align*}
Define
\begin{align*}
&I_5(n,k,h) \\
&=\int_{\{|u_n-T_k(u)|<h\}} \big(\hat{a}(x,T_n(u_n),\nabla u_n)
-\hat{a}(x,T_n(u_n),\nabla T_k(u))\big)\nabla T_{h}( u_n-T_k(u))\,dx,
\end{align*}
which we split as the difference $I_6-I_7$, where
\begin{gather*}
I_6(n,k,h)=\int_{\{|u_n-T_k(u)|<h\}} \hat{a}(x,T_n(u_n),
\nabla u_n)\nabla T_{h}(u_n-T_k(u))\,dx, \\
I_7(n,k,h)=\int_{\{|u_n-T_k(u)|<h\}} \hat{a}(x,T_n(u_n),
 \nabla T_k(u))\nabla T_{h}(u_n-T_k(u))\,dx.
\end{gather*}
Take $n$ sufficiently large such that $n>h+k$. Since 
$| u_n|\leq k + h$ on the set where 
$\left\{ | u_n-T_k( u)| \leq h\right\}$, we obtain
$$
I_7(n,k,h)=\int_{\Omega} \hat{a}(x,T_{h+k}(u_n),
\nabla T_k(u))\nabla T_{h}(u_n-T_k(u))\,dx.
$$
According to  condition \eqref{a1-condition}, we have
$$
|\hat{a}_i(x,T_{h+k}(u_n),\nabla T_k(u))|
\leq \beta|D_iT_k(u))|^{p_i(x)-1},\ \forall i=1,\dots ,N. 
$$
Note that
$$
\hat{a}_i(x,T_{h+k}(u_n),\nabla T_k(u))\to \hat{a}_i(x,u,\nabla T_k(u))\quad
\text{a.e. in } \Omega, \; i=1,\dots ,N,
$$
using Lebesgue dominated convergence theorem, we derive
$$
\hat{a}_i(x,T_{h+k}(u_n),\nabla T_k(u))\to \hat{a}_i(x,u,\nabla T_k(u))\quad
 \text{strongly in } L^{p_i'(\cdot)}(\Omega), \forall i=1,\dots ,N.
$$
Using the weak convergence of $D_i T_{h}(u_n-T_k(u))$ to $D_i T_{h}(u-T_k(u))$ in
$L^{p_i(\cdot)}(\Omega), i=1,\dots ,N$ (a consequence of (i) and (ii)), we find
$$
\lim_{n\to+\infty}I_7(n,k,h)=\int_{\Omega} 
\sum_{i=1}^{N}(\hat{a}_i(x,u, \nabla T_k( u))D_i T_{h}(u-T_k(u)))\,dx,
$$
so
\begin{equation}\label{20}
\lim_{k\to +\infty}\lim_{n\to +\infty}I_7(n,k,h)=0.
\end{equation}
For $I_6(n,k,h)$, by \eqref{p1} we obtain
$$
I_6(n,k,h)=\int_{\Omega} f_nT_{h}(u_n-T_k(u))\,dx,
$$
by the strong convergence of $f_n$ in $L^{1}(\Omega)$, we have
\begin{equation}\label{21}
\lim_{k\to+\infty}\lim_{n\to+\infty}I_6(n,k,h)=0
\end{equation}
Putting together \eqref{18}, \eqref{19}, \eqref{20}, and \eqref{21}, one thus has
$$
\lim_{n\to+\infty}I(n)=0.
$$
As in  \cite{xi}, we obtain
$D_i u_n\to D_i u$ a.e. in $\Omega$, $i=1,\dots ,N$.
\end{proof}

\begin{lemma}\label{lower}
Let $u_n$ be a solution to the equation \eqref{approximate-equation}, 
suppose that $u_n$ converges to $u$ almost everywhere in $\Omega$. Then
$$
|u_n|^{s(x)-1}u_n\to |u|^{s(x)-1}u\ \text{in}\ L^{1}(\Omega).
$$
\end{lemma}

\begin{proof}
 Let $\rho_{j}(t)$ be an increasing, uniformly bounded Lipschitz function 
such that $\rho_{j} \to \chi_{\{|t|>k\}}\operatorname{sign}(t)$ $(k>0)$, as 
$j\to +\infty$. Taking $\rho_{j}(u_n)$ as a test function in
\eqref{app-variationel}, we obtain
\begin{align*}
&\sum^{N}_{i=1}\int_{\Omega}\rho'_{j}(u_n) a_i(x,T_n(u_n))| D_iu_n|^{p_i(x)}\,dx
+\int_{\Omega}| u_n|^{s(x)-1}u_n\rho_{j}(u_n) \,dx \\
&=\int_{\Omega}T_n(f) \rho_{j}(u_n) \,dx.
\end{align*}
As $j\to+\infty$, we obtain
\begin{equation}\label{22}
\int_{\{| u_n|>k\}}| u_n|^{s(x)}\,dx\leq \int_{\{| u_n|>k\}}| f| \,dx.
\end{equation}
Let $E \subset \Omega$ be any measurable set, using \eqref{22}. We have
\begin{align*}
 \int_{E}| u_n|^{s(x)}\,dx
&= \int_{E\cap\{| u_n|\leq k\}}| u_n|^{s(x)}\,dx
 +\int_{E\cap\{| u_n|>k\}}| u_n|^{s(x)}\,dx\\
&\leq  (k^{s^{+}}+k^{s^{-}})\operatorname{meas}(E)+\int_{E\cap\{| u_n|>k\}}| f| \,dx.
\end{align*}
Then we deduce that $(|u_n|^{s(x)-1}u_n)$ is equi-integrable in $L^{1}(\Omega)$, 
and since $u_n\to u\ \text{a.e. in}\ \Omega$, then Vitali's theorem implies 
\begin{equation}\label{convL1}
 | u_n|^{s(x)-1}u_n\to| u|^{s(x)-1}u\quad \text{in } L^{1}(\Omega).
\end{equation}
\end{proof}

\section{Proof of main results}

In this section, using the uniform estimates of Section \ref{sec4}, 
we prove Theorem \ref{thm1}, \ref{thm2}, \ref{thm3}
and \ref{thm4}.

\subsection{Proof of theorems \ref{thm1}, \ref{thm2}}
By Lemma \ref{lemma-3} the sequence $(u_n)$ is bounded in
 $ \mathring{W}^{1,\vec{q}(\cdot)}(\Omega)$ where $q_i(\cdot)$ is 
defined as \eqref{regularite-th2}. Without loss of generality, we can 
therefore assume that
\begin{equation} \label{conv}
\begin{gathered}
 u_n\rightharpoonup u \quad \text{weakly in } \mathring{W}^{1,\vec{q}(\cdot)}(\Omega), \\
 u_n\to u \quad \text{strongly in }  L^{q_0}(\Omega),  \quad
 q_0=\min_{1\leq i\leq N}\min_{x\in\overline{\Omega}}q_i(x), \\
 u_n\to u \quad \text{a.e. in}\quad\Omega.
\end{gathered}
\end{equation}
It follows from \eqref{est1} and Fatou's lemma that
\begin{equation*}
 \int_{\Omega}|u|^{s(x)}\,dx\leq \liminf_{n\to+\infty}
\int_{\Omega}|u_n|^{s(x)}\,dx\leq C,
\end{equation*}
thus $|u|^{s(x)}\in L^{1}(\Omega)$, furthermore, $u\in \mathcal{M}^{s^{-}}(\Omega)$.
Then, there exists $r_1=s^{-} > 0$ such that
\begin{equation}\label{bond in marsi}
 \|u_n\|_{M^{r_1} (\Omega)}\leq C \quad\text{and}\quad u\in M^{r_1} (\Omega).
\end{equation}
Let $f_n=T_n(f)-T_n(|u_n|^{s(x)-1}u_n)\in L^{\infty}(\Omega)$, 
where $u_n$ is a solution of \eqref{p1}.
Then, from \eqref{tranestimate}, \eqref{est3}, \eqref{bond in marsi}, 
\eqref{conv}, and lemma \ref{gradient} we can deduce that
$$
D_iu_n\to D_iu \quad \text{a.e. in } \Omega,  \text{for all } i=1,\dots ,N.
$$
So, by \eqref{est3}, we have
\begin{equation}\label{pas}
 |D_iu_n|^{p_i(x)-2}D_iu_n\rightharpoonup |D_iu|^{p_i(x)-2}D_iu \quad
 \text{weakly in}\ L^{\frac{q_i(\cdot)}{p_i(\cdot)-1}}(\Omega),\;
 \forall i=1,\dots ,N,
\end{equation}
where $q_i$ is defined as in \eqref{regularite-th2}. 
The choice of $\frac{q_i(\cdot)}{p_i(\cdot)-1}>1$ is possible since we 
have \eqref{cond-th2-p}.
From \eqref{a1-condition} and \eqref{conv}, we obtain
\begin{equation}\label{con*}
 a_i(x,T_n(u_n))\to a_i(x,u)\quad\text{weak$^*$ in } L^\infty(\Omega).
\end{equation}
For any given $\varphi\in C^\infty_0(\Omega)$, using $\varphi$ 
as a test  function in \eqref{approximate-equation}, we have
\begin{equation}\label{e101}
\sum^{N}_{i=1}\int_{\Omega} a_i(x,T_n(u_n))| D_iu_n|^{p_i(x)-2}D_iu_nD_i\varphi \,dx
+\int_{\Omega}| u_n|^{s(x)-1}u_n\varphi \,dx=\int_{\Omega}T_n(f) \varphi \,dx,
\end{equation}
Letting $n\to+\infty$ in \eqref{e101}, by \eqref{pas}, \eqref{con*}, and 
\eqref{convL1}, we obtain
$$
\sum^{N}_{i=1}\int_{\Omega} a_i(x,u)| D_iu|^{p_i(x)-2}D_iuD_i\varphi \,dx
+\int_{\Omega}| u|^{s(x)-1}u\varphi \,dx=\int_{\Omega}f \varphi \,dx.
$$
For the proof of Theorem \ref{thm2}, we only replace \eqref{pas} with 
 $$
|D_iu_n|^{p_i(x)-2}D_iu_n\rightharpoonup |D_iu|^{p_i(x)-2}D_iu \quad
 \text{weakly in } L^{\frac{q_i(\cdot)}{p_i(\cdot)-1}}(\Omega),\;
 \forall i=1,\dots ,N.
$$
where $q_i$ is defined as in \eqref{regularite-1}. The choice of 
$\frac{q_i(\cdot)}{p_i(\cdot)-1}>1$ is possible since we have \eqref{s-cond-th1}.

\subsection{Proof of theorem \ref{thm3}, \ref{thm4}}
Because the proof of Theorem \ref{thm4} is similar to that of
Theorem \ref{thm1}, here we only give the proof of Theorem \ref{thm3}.
 According to Lemma \ref{lemma-7}, the sequence $(u_n)$ is bounded in 
$ \mathring{W}^{1,\vec{p}(\cdot)}(\Omega)$. This implies that we can extract a 
subsequence (denote again by $(u_n)$), such that
\begin{gather*}
 u_n\rightharpoonup u \quad \text{weakly in } \mathring{W}^{1,\vec{p}(\cdot)}(\Omega),\\
 u_n\to u \quad \text{strongly in }  L^{p_0}(\Omega), \quad
 p_0=\min_{1\leq i\leq N}\min_{x\in\overline{\Omega}}p_i(x),\\
 u_n\to u \quad \text{a.e. in } \Omega.
\end{gather*}
Arguing as the proof of Theorem \ref{thm1}, by using \eqref{est-lemma-7},
 we conclude that
\begin{equation*}
 |D_iu_n|^{p_i(x)-2}D_iu_n\rightharpoonup |D_iu|^{p_i(x)-2}D_iu \quad
 \text{weakly in } L^{p'_i(\cdot)}(\Omega),\; \forall i=1,\dots ,N.
\end{equation*}
The proof of Theorem \ref{thm3} is complete.

\begin{remark} \label{rmk6.1} \rm
All the results in this work also hold if our problem is exchanged by a more 
general one,
\begin{gather*}
 -\operatorname{div}(a(x,u,\nabla u))+g(x,u)=f \quad \text{in }\Omega, \\
 u=0 \quad \text{on }\partial \Omega,
 \end{gather*}
where $a(x,t,\xi)=\{ a_i(x,t,\xi)\}_{i=1,\dots ,N}:
\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}^{N}$
is a Carath\'eodory vector-valued function such that for a.e. 
$x\in\Omega$ and for every $(t,\xi) \in \mathbb{R} \times\mathbb{R}^{N}$, 
the following assumptions hold:
\begin{gather*}
a(x,t,\xi)\xi\geq\alpha\sum_{i=1}^{N} 
\frac{|\xi_i|^{p_i(x)}}{(1+|t|)^{\gamma_i(x)}},\quad \alpha>0, \\
| a_i(x,t,\xi)| \leq \beta\Big(1+\sum_{j=1}^{N}|\xi_{j}|^{p_{j}(x)}
 \Big) ^{1-\frac{1}{p_i(x)}},\quad i=1,\dots ,N,\;\beta > 0,\\
(a(x,s,\xi)-a(x,s,\xi'))(\xi-\xi')>0,\quad \forall \xi\neq\xi'.
\end{gather*}
Assume that $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'eodory 
function satisfying
\begin{gather*}
\sup_{|t|\leq k}|g(x,t)|=h_k(x)\in L^{1}(\Omega),\quad \forall k>0, \\
 g(x,t)\operatorname{sign}(t)\geq |t|^{s(x)}.
\end{gather*}
\end{remark}

\subsection*{Acknowledgements}
The authors would like to thank the referees for their comments and suggestions. 
They are also grateful to Mr. Lakhdar Benaissa, the head of the department 
of Mathematics and Informatics at Benyoucef Benkhedda University, 
who provided all the means to the research team.


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\end{document}