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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 163, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/163\hfil Existence and nonexistence of solutions]
{Existence and nonexistence of solutions to  nonlinear gradient
elliptic systems involving  $(p(x),q(x))$-Laplacian operators}

\author[O. Saifia, J. V\'elin \hfil EJDE-2014/163\hfilneg]
{Ouarda Saifia, Jean V\'elin}  % in alphabetical order

\address{Ouarda Saifia \newline
Department of mathematics,
University of Annaba, PO 12, 
El Hadjar, 23000,  Annaba, Algeria}
\email{wsaifia@gmail.com}

\address{Jean V\'elin \newline
Department of Mathematics and Computers,
Laboratory CEREGMIA, University of Antilles-Guyane,
Campus de Fouillole,  97159 Pointe-\`a-Pitre , Guadeloupe (French West Indies)}
\email{jean.velin@univ-ag.fr}


\thanks{Submitted April 2, 2014. Published July 25, 2014.}
\subjclass[2000]{35J20, 35J35, 35J45, 35J50, 35J60, 35J70}
\keywords{Fibering method;  $p(x)$-Laplacian;
  Generalized Pohozeav identity; \hfill\break\indent  Pucci-Serrin identity}

\begin{abstract}
 In this article, we establish the existence of nontrivial solutions
 by employing the fibering method introduced by  Pohozaev.
 We also generalize the well-known Pohozaev and Pucci-Serrin identities
 to a $(p(x),q(x))$-Laplacian system. A nonexistence result
 for a such system is then proved.
\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}

After the pioneer work by Kovacik and Rokosnik \cite{kr} 
concerning the $ L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$ spaces, 
many researches have studied the variable exponent spaces.
We refer to \cite{xjd} for the properties of such spaces and \cite{cf,g} 
for the applications of variable exponent on partial differential equations.
In the recent years,  problems  with $p(x)$-Laplacian have been applied to a large
number of application in nonlinear electrorheological fluids, elastic mechanics,  
image processing, and flow in porous media 
(see for instance \cite{am,as,clr,d,h,mr,r,z}).

In this article,  we study the existence and non-existence of the weak 
solutions for the following $(p(x),q(x))$-gradient elliptic system:
\begin{equation}\label{2}
\begin{gathered}
- \Delta _{p(x)}u=c(x)u |u|^{\alpha-1} |v|^{\beta+1} \quad \text{in } \Omega  \\
- \Delta _{q(x)}v=c(x)v |v|^{\beta-1} |u|^{\alpha+1} \quad \text{in } \Omega  \\
u=v=0 \quad \text{on } \Omega.
\end{gathered}
\end{equation}
Here $\Omega$ designates a bounded and open set in $\mathbb{R}^{N} $,
with a smooth boundary $\partial\Omega$.
$ p,q :\Omega  \to  \mathbb{R}$ are  two measurable  functions  
from ${\Omega}$ to $[1,+\infty)$, and   $c$ is a function with  changing sign. 
Concerning  the existence and nonexistence results for such systems, 
 we cite the work \cite{bm}. There the authors use the fibering method introduced
 by  Pohozeav. They obtained the existence of multiple solutions for a 
Dirichlet problem associated with a quasilinear system involving a pair of 
$ (p,q)$-Laplacian operators.
 Recently, Velin \cite{v1,v2}, employing the fibering method,  
 proved the existence of multiple positive solutions for a class of 
$(p,q)$-gradient elliptic systems including systems like \eqref{2}.

Systems structured as \eqref{2} have been investigated for instance
in \cite{tv}.  There the authors  presented some results dealing with  existence
and nonexistence of a non-trivial solution $(u,v)\in W_{0}^{1,p}(\Omega)\times
W_{0}^{1,q}(\Omega)$ of the  system
\begin{equation}  \label{a}
\begin{gathered}
- \Delta _{p}u=u |u|^{\alpha-1} |v|^{\beta+1} \quad \text{in } \Omega  \\
- \Delta _{q}v=v |v|^{\beta-1} |u|^{\alpha+1} \quad \text{in}\; \Omega  \\
u=v=0 \quad \text{on }\Omega.
\end{gathered}
\end{equation}
The authors have proved nonexistence results
when $\Omega$  is a strictly starshaped open domain in $\mathbb{R}^{N}$ and  
 \begin{equation}\label{pqcondition1}
(\alpha+1)\frac{N-p}{Np}+(\beta+1)\frac{N-q}{Nq}\geq 1\,.
\end{equation}
 On the other hand, under the  assumptions 
\begin{equation}\label{pqcond2}
(\alpha+1)\frac{N-p}{Np}+(\beta+1)\frac{N-q}{Nq}<1,\quad
\frac{\alpha+1}{p}+\frac{\beta+1}{q}\neq 1, 
\end{equation}
 some existence results have been obtained.
In \cite{di}, the authors deal with nonexistence for an elliptic Dirichlet
equation governed by the $p(x)$-Laplacian operator.


 The article has the following structure.
Section 2 is devoted to introduce some notation and preliminaries needed 
for the framework of the paper. We also recall some tools defined by the theory 
of variable  exponents Lebesgue and Sobolev spaces.
Section 3 states the main results.
In Section 4, following the ideas explained in \cite{di},   we establish a
Pohozaev-type identity for the system \eqref{2}. By using this  identity,
 we deal with the non-existence results of non trivial solutions.
 In section 5, after recalling the spirit of the fibering method,  
we show  that \eqref{2} admits at least one weak non-trivial solution.

\section{Preliminaries}

Let ${\mathcal P}(\Omega)$ denote the set  
$\{p; p:\Omega\to [1,+\infty) \text{ is measurable } \}$.
$\Omega\subset \mathbb{R}^{N}$ is an open set. 
$L^{p(x)}(\Omega)$  designates the generalized Lebesgue space.  
$L^{p(x)}(\Omega)$ consists of all measurable functions $u$ defined on 
$\Omega$ for which the $p(x)$-modular
\begin{equation*}
\rho _{p(.)}(u)=\int_{\Omega }| u(x)| ^{p(x)}dx
\end{equation*}
is finite. The Luxemberg norm on this space is defined as
\begin{equation*}
\| u\|_{p} =\inf \{\lambda >0;\rho _{p(.)}(u)=\int_{\Omega
}| \frac{u(x)}{\lambda }| ^{p(x)}dx\leq 1\}.
\end{equation*}
Equipped with this norm, $L^{p(x)}(\Omega)$  is a Banach space.
 Some basic results on the generalized Lebesgue spaces can be find in 
  \cite{dhhr,xd,f,g,OS,GS,kr,mr,m}.
If $p(x)$ is constant, $L^{p(x)}(\Omega)$ is reduced to the standard Lebesgue space.

For any  $p \in {\mathcal P}(\Omega)$ and $ m \in \mathbb{N}^*$, the generalized 
Sobolev space $W^{m,p(x)}(\Omega)$ is defined by
\begin{gather*}
W^{m,p(.)}(\Omega )=\{u\in L^{p(.)}(\Omega ):D^{\alpha }u\in
L^{p(.)}(\Omega ) \text{ for all }| \alpha | \leq m\}, \\
\| u\| _{m,p(.)}=\sum_{| \alpha | \leq
m}\| D^{\alpha }u\| _{L^{p(.)}(\Omega )}\,.
\end{gather*}

The pair $(W^{m,p(.)}(\Omega ),\| \cdot\| _{m,p(.)}) $ is a separable Banach 
space (reflexive if $ p^{-}>1$).
 $ W_{0}^{1,p(.)}(\Omega)$ denotes the closure of
   $ C_{0}^{\infty}(\Omega)$ in $ W^{1,p(.)}(\Omega)$.
On the generalized Sobolev space, we refer to the works due to
 \cite{er, xjd, xd,  xzg, hhkv, hh, kr}.

 We define:
$p,q:\Omega \to [1,+\infty)$ as two measurable functions.

For a given measurable function $p:\Omega\to [1,+\infty)$, the conjugate 
function designated by
$$ 
p'(x)=\frac{p(x)}{p(x)-1}.
 $$

A function $ p: \Omega \to \mathbb{R}$ is $\ln$-H\"older continuous on $\Omega$ 
(See \cite{xd}),
provided that there exists a constant $ L>0$ such that
\begin{equation}\label{polder}
| p(x)-p(y)| \leq \frac{L}{-\ln| x-y| },\quad  \text{for all }  x,y\in \Omega,\;
| x-y| \leq \frac{1}{2}.
\end{equation}

\begin{gather*}
p^{-}=\min_{x\in\overline{\Omega}}p(x), \quad
q^{-}=\min_{x\in\overline{\Omega}}q(x),  \\
p^{+}=\max_{x\in\overline{\Omega}}p(x), \quad
q^{+}=\max_{x\in\overline{\Omega}}q(x).
\end{gather*}

For  $c:\Omega\to \mathbb{I}$,  $c_+(x)\neq 0$, $c_-(x)\neq 0$.


\section{Main results}


Let us now state the main results of this paper:
\subsection*{A non-existence result for the $(p(x),q(x))$-Laplacian system \eqref{2}}

\begin{theorem}\label{NE}
 Let $\Omega$ be a bounded open set of
$\mathbb{R}^{N}$, with boundary $\partial\Omega$ of  class $C^{1}$. 
Let 
  $p,q:\Omega\to\mathbb{I}$ functions of class 
$C_{B}^{1}(\Omega)\cap\;C(\overline{\Omega})$, $p^{-},q^{-}>1$,
and 
$c(.)\in C_B^1(\Omega\setminus{\mathcal C})$, with 
$\operatorname{meas}({\mathcal C})=0$. 
Assume that
 $\Omega$ be a  bounded domain of class $C^{1}$, starshaped
with respect to the origin;
$(p,q)\in C_{B}^{1}(\Omega)\cap C(\bar{\Omega})$; 
$p^{-},q^{-} >1$; and 
$(x\cdot\nabla p) \geq 0$, $(x\cdot\nabla q) \geq 0$,
\begin{gather}\label{gradc}
\langle x,\nabla c(x)\rangle \leq 0 \quad\text{ for any $x$ in }\Omega,\\
\label{Np+}
(\alpha+1) \frac{N-p^+}{Np^+}+ (\beta+1)\frac{N-q^+}{Nq^+}\geq 1.
\end{gather}
Then \eqref{2} has not a nontrivial classical
solution $(u,v)\in (C^{2}(\Omega)\cap C^{1}(\bar {\Omega}))^2$
which satisfies:
\begin{equation}  \label{uvepq}
|\nabla u(x)|\geq e^{1/p(x)},\quad  |\nabla
v(x)|\geq e^{1/q(x)}\quad \text{a.e } x\in\Omega,
\end{equation}
and
$$
\int_\Omega c(x)|u|^{\alpha+1}|v|^{\beta+1}dx >0.
$$
\end{theorem}

\subsection*{An existence result for the $(p(x),q(x))$-Laplacian system \eqref{2} }

\begin{theorem}\label{E}
 Let $\Omega$ be a bounded open set of
$\mathbb{R}^{N}$, with boundary $\partial\Omega$ of  class $C^{1}$. Let   
$p,q:\Omega\to\mathbb{I}_+^*$ two functions of class 
$C_{B}^{1}(\Omega)\cap\;C(\overline{\Omega})$; $p_{-},q_{-}>1$.
Assume that:
\begin{gather}  \label{Np-}
 (\alpha+1) \frac{N-p^-}{Np^-}+ (\beta+1)\frac{N-q^-}{Nq^-}< 1,\\
\label{alfa+}
\gamma^+=\frac{\alpha+1}{p^{+}}+\frac{\beta+1}{q^{+}}-1>0.
\end{gather}
Then system \eqref{2}  admits at least one nontrivial solution 
$(u^*,v^*)\in W_{0}^{1,p(x)}(\Omega)\times W_{0}^{1,q(x)}(\Omega)$.
Moreover,  one have
\begin{gather*}
\|u^*\|_{1,p(x)}^{p^+}=\|v^*\|_{1,q(x)}^{q^+}, \\
\int_\Omega c(x)|u^*|^{\alpha+1}|v^*|^{\beta+1}dx >0.
\end{gather*}
\end{theorem}

\begin{remark} \rm
Let us remark that conditions \eqref{Np+} and \eqref{Np-} seem to generalize 
to $(p(x),q(x))-$ gradient elliptic systems  conditions \eqref{pqcondition1} 
and \eqref{pqcond2} well known when $(p,q)-$ gradient elliptic systems are 
considered. Obviously,   conditions \eqref{Np+} and \eqref{Np-}
imply respectively 
\begin{gather*}
1\leq (\alpha+1) \frac{N-p^-}{Np^-}+ (\beta+1)\frac{N-q^-}{Nq^-},\\
(\alpha+1) \frac{N-p^+}{Np^+}+ (\beta+1)\frac{N-q^+}{Nq^+}< 1.
\end{gather*}
\end{remark}

\section{A Pohozaev-type identity for $(p(x),q(x))$-Laplacian and a nonexistence
result}

Consider the elliptic system with Dirichlet boundary condition:
\begin{gather*}
-\Delta_{p(x)}u=c(x) u|u|^{\alpha-1}|v|^{\beta+1} \quad \text{in } \Omega  \\
-\Delta_{q(x)}v=c(x)|u|^{\alpha+1}v|v|^{\beta-1}\quad \text{in }\Omega  \\
u=v=0 \quad \text{on } \Omega,
\end{gather*}
where $\Omega\subset\mathbb{I}^{N}$ is a bounded open set with a regular  boundary 
$\partial\Omega$;   $p,q, c$ are defined as in the previous section.
\[
 \Delta_{p(x)}u=\frac{\partial } {\partial x_{i}} \Big(|\nabla u|^{p(x)-2}
\frac{\partial u} {\partial x_{i}}\Big).
\]

\begin{proposition}\label{pro1}
Let $\Omega$ be a bounded open set of
$\mathbb{R}^{N}$, with boundary $\partial\Omega$ of  class $C^{1}$. Assume that
 $p,q:\Omega\to\mathbb{I}$ are two functions of class 
$C_{B}^{1}(\Omega)\cap\;C(\overline{\Omega})$;
 $p^{-},q^{-}>1$;
$c(.)\in C_B^1(\Omega\setminus{\mathcal C})$, with 
$\operatorname{meas}({\mathcal C})=0$ and  
\begin{equation*}
\langle x,\nabla c(x)\rangle \leq 0 \quad \text{for any $x$ in }\Omega.
\end{equation*}
For every classical solution 
$(u,v)\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$ 
of \eqref{2}, the following identity holds:
\begin{align*}
&\frac{\alpha+1}{N} \int_{\partial \Omega} \frac{1-p(x)}{p(x)}
|\nabla u|^{p(x)} \langle x,\nu\rangle {\mathrm{d}}\sigma +\frac{\beta+1}{N}
\int_{\partial \Omega} \frac{1-q(x)}{q(x)}|\nabla v|^{q(x)}
\langle x,\nu\rangle {\mathrm{d}} \sigma \\
&=\frac{\alpha+1}{N} \int_{\Omega} \big(\frac{N-p(x)}{p(x)}
-a_1\big)|\nabla u|^{p(x)}{\mathrm{d}}x +\frac{\beta+1}{N}
\int_{\Omega} \big(\frac{N-q(x)}{q(x)}-a_2\big)|\nabla v|^{q(x)}{
\mathrm{d}}x  \\
& \quad +\int_{\Omega} \Big[\frac{1}{p^{2}(x)} 
 \langle x\cdot\nabla p\rangle (\ln
|\nabla u|^{p(x)}-1)|\nabla u|^{p(x)}  \\
&\quad + \frac{1}{q^{2}(x)} \langle x, \nabla q\rangle (\ln
|\nabla v|^{q(x)}-1)|\nabla v|^{q(x)} \Big]{\mathrm{d}}x \\
&\quad +\int_{\Omega}\big\{(\alpha+1)a_1+(\beta+1)a_2-N\big\}
 c(x)|u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x\\
&\quad -\int_{\Omega}\langle x,\nabla c\rangle|u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x.
\end{align*}
for all $a_1$ and $a_2\in \mathbb{R}^{N}$.
\end{proposition}

Before proving the proposition \ref{pro1}, we present the following result
generalizing the variational identity of {Pucci-Serrin} \cite{ps}.

\begin{proposition}\label{pro2}
Let $\Omega$ be a bounded open set of
$\mathbb{R}^{N}$  with boundary $\partial\Omega$ of  class $C^{1}$. 
Assume that $ p,q:\Omega\to\mathbb{I}$ are tow functions of class
$C_{B}^{1}(\Omega)\cap C(\overline{\Omega})$; 
$p^{-},q^{-}>1$;
$c(\cdot)\in C_B^1(\Omega\setminus{\mathcal C})$, 
${\mathcal C}\subset\Omega$ with $\operatorname{meas}({\mathcal C})=0$.
 For every classical solution 
$(u,v)\in \big( C^{2}(\Omega)\cap C^{1}(\overline{\Omega})\big)^2$ of
problem \eqref{2},  the following equality  holds
\begin{equation}  \label{4}
\begin{aligned}
&\frac{\partial}{\partial x_{i}} \Big[x_{i}\Big(\frac{\alpha+1}{p(x)}
|\nabla u|^{p(x)}+ \frac{\beta+1}{q(x)}|\nabla v|^{q(x)}-
c(x)|u|^{\alpha+1}|v|^{\beta+1}\Big) \\
&-  (\alpha+1)\big(x_{j}\frac{\partial u}{
\partial x_{j}}+ a_1u\big)|\nabla u|^{p(x)-2}
\frac{\partial u} {\partial x_{i}} -(\beta+1)\Big(x_{j}\frac{\partial v}{
\partial x_{j}}+a_2v\Big)|\nabla v|^{q(x)-2}\frac{
\partial v}{\partial x_{i}} \Big] \\
&=(\alpha+1)\Big[\frac{N-p(x)}{p(x)}-a_1 \Big]|\nabla u|^{p(x)} +
(\beta+1)\Big[\frac{N-q(x)}{q(x)}-a_2 \Big]|\nabla v|^{q(x)} \\
&\quad +\frac{\langle x,\nabla p\rangle}{p^{2}(x)} 
 (\ln |\nabla u|^{p(x)}-1)|\nabla u|^{p(x)} 
 + \frac{\langle x,\nabla q\rangle}{q^{2}(x)} 
 (\ln |\nabla v|^{q(x)}-1)|\nabla v|^{q(x)} \\
&\quad +\{(\alpha+1)a_1+(\beta+1)a_2-N\}c(x)|u|^{\alpha+1}|v|^{\beta+1}
 -\langle x, \nabla c\rangle |u|^{\alpha+1}|v|^{\beta+1} ,
\end{aligned}
\end{equation}
for all $a_1$ and $a_2$ in $\mathbb{R}$.
\end{proposition}

The proof of Proposition \ref{pro2} can be established by a simple computation.


\begin{proof}[Proof of Proposition \ref{pro1}]
In this proof, for any vectors in $\mathbb{I}^N$ 
$x=(x_i)_{i=1,\dots,N}$ and  
$y=(y_i)_{i=1,\dots,N}$, the classical inner product  $xy$ is  denoted 
$x_iy_i$ and the notation $\sum_{i=1}^N$ is omitted.
Let $(u,v) \in \left(C^{2}_{B} \cap C^{1} (\bar\Omega)\right)^2$ be a classical 
solution of the problem \eqref{2}. According to the {Proposition} \ref{pro2}, 
$(u,v)$ satisfies the identity \eqref{4}. Integrating by part  over $\Omega$, 
we get
\begin{equation} \label{5}
\begin{aligned}
&\int_{\partial \Omega}\Big[\Big(\frac{\alpha+1}{
p(x)}|\nabla u|^{p(x)} + \frac{\beta+1}{q(x)}|\nabla
v|^{q(x)}-c(x)|u|^{\alpha+1}|v|^{\beta+1} \Big) \\
& -(\alpha+1)\big(x_{j}\frac{\partial u}{\partial x_{j}}+a_1u\big)|\nabla u|^{p(x)-2}
 \frac{\partial u}{\partial x_{i}}\\
& -(\beta+1)\big(x_{j}\frac{\partial v}{\partial x_{j}}+a_2v\big)|\nabla v|^{q(x)-2}
 \frac{\partial v}{\partial x_{i}} \Big] \nu_{i} {\mathrm{d}}\sigma\\
& =(\alpha+1)\int_{\Omega}\big(\frac{N-p(x)}{p(x)}-a_1
\big)|\nabla u|^{p(x)}{\mathrm{d}}x \\
&\quad +(\beta+1) \int_{\Omega}\big(\frac{N-q(x)}{q(x)}-a_2 \big)
 |\nabla v|^{q(x)}{\mathrm{d}}x  \\
&\quad +\int_{\Omega} \Big[\frac{1}{p^{2}(x)} \langle x\cdot\nabla p\rangle 
(\ln|\nabla u|^{p(x)}-1)|\nabla u|^{p(x)}\\ 
&\quad + \frac{1}{q^{2}(x)} \langle x, \nabla q\rangle (\ln
|\nabla v|^{q(x)}-1)|\nabla v|^{q(x)} \Big]{\mathrm{d}}x\\
&\quad +\int_{\Omega}\Big\{(\alpha+1)a_1+(\beta+1)a_2-N\Big\}c(x)|u|^{\alpha+1}
 |v|^{\beta+1}{\mathrm{d}}x\\
&\quad -\int_{\Omega}\langle x,\nabla c\rangle|u|^{\alpha+1}
|v|^{\beta+1}{\mathrm{d}}x,
\end{aligned}
\end{equation}
where $\nu$ is
the unit outer normal to the boundary $\partial \Omega$
Since $u=0 $ on $\partial \Omega$,  clearly it follows  that
$\frac{\partial u} {\partial {x_{i}}}=(\nabla u.\nu)\nu_{i}$
for $i=1,\dots,N$. 
Then, for $x$  on  $\partial \Omega$, we can write
\begin{align*}
x_{j} \frac{\partial u} {\partial x_{j}} \frac{\partial u} {\partial x_{i}}
|\nabla u|^{p(x)-2}\nu _{i}
&= x_{j}[(\nabla u.\nu)\nu_{j}]\frac{\partial u} {\partial x_{i}}
 |\nabla u|^{p(x)-2}\nu_i \\
&=\frac{\partial u} {\partial x_{i}}\frac{\partial u} {\partial x_{i}}
|\nabla u|^{p(x)-2}(x.\nu) \\
&=|\nabla u|^{p(x)}(x.\nu)\quad \text{on }\partial \Omega
\end{align*}
Using the relation \eqref{5} and the fact that $u|_{\partial \Omega}=0$ in
the left hand side of this relation, the statement of the 
Proposition \ref{pro1} occurs.
\end{proof}


\begin{remark}\label{cexemple}\rm
Before proving Proposition \ref{pro2}, we note that the set of functions $c$ 
satisfying to hypothesis \eqref{gradc}, is non-empty. Indeed, let  $x_0$ be 
in $\partial\Omega$ such that $\operatorname{dist}(0,\partial\Omega)=\operatorname{dist}(0,x_0)$. 
We set $R_0=\operatorname{dist}(0,\partial\Omega)$. Obviously, we remark that the 
ball $B(0,R_0)$ is contained in $\Omega$. We define the set $\Omega_1$ 
by $\Omega_1=\{x\in\Omega;  0\leq \|x\|\leq R_0/2 \}$. 
For instance, we define the function
\[
c(x)=\begin{cases}
-e^{\| x\|^2}&\text{if }x\in\Omega_1\\
e^{-\| x\|^2}&\text{if }x\in\Omega\setminus\Omega_1.
\end{cases}
\]
This function changes sign in $\Omega$ and we also have for any 
$x\in\Omega$,  $\langle x, \nabla c(x)\rangle\leq 0$. Moreover,
 $c\in L^{\infty}(\Omega)$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{NE}]
Suppose that there exists a nontrivial classical solution 
$(u,v)$ in $C^{2}(\Omega)\cap C^{1}(\bar {\Omega})$ of the problem \eqref{2}. 
So that,  $(u,v)$ satisfies the statement of Proposition \ref{pro1}.
Since $\Omega \subset \mathbb{R}^{N} $ is strictly starshaped with respect to
the origin, we have $x\cdot\nu >0$ on $\partial \Omega $ thus
\begin{equation*}
-\frac{\alpha+1}{N} \int_{\partial \Omega}\frac{1}{{\tilde p}
(x)} |\nabla u|^{p(x)} \langle x,\nu \rangle {\mathrm{d}}\sigma -\frac{\beta+1}{N}
\int_{\partial \Omega} \frac{1}{{\tilde q}(x)}|\nabla
v|^{q(x)} \langle x,\nu \rangle {\mathrm{d}} \sigma <0,
\end{equation*}
where $\frac{1}{{\tilde p}(x)}=\frac{p(x)-1}{p(x)}$, 
$\frac{1}{{\tilde q} (x)}=\frac{q(x)-1}{q(x)}$.

On other hand, choosing $a_1\in \mathbb{I}$ and $a_2\in\mathbb{I}$ such that 
$$
(\alpha+1)\frac{a_1}N+ (\beta+1)\frac{a_2}N= 1
$$   
and using the relations \eqref{Np+}, \eqref{uvepq}, we obtain
\begin{align*}
&\frac{\alpha+1}{N} \int_{\partial \Omega} \frac{1-p(x)}{p(x)}
|\nabla u|^{p(x)} \langle x,\nu\rangle {\mathrm{d}}\sigma +\frac{\beta+1}{N}
\int_{\partial \Omega} \frac{1-q(x)} 
{q(x)}|\nabla v|^{q(x)} \langle x,\nu\rangle {\mathrm{d}} \sigma \\ 
&=\frac{\alpha+1}{N} \int_{\Omega} \Big(\frac{N-p(x)}{p(x)}
-a_1\Big)|\nabla u|^{p(x)}{\mathrm{d}}x +\frac{\beta+1}{N}
\int_{\Omega} \Big(\frac{N-q(x)}{q(x)}-a_2\Big)|\nabla v|^{q(x)}{
\mathrm{d}}x\\
&\quad +\int_{\Omega}\Big\{(\alpha+1)a_1
 +(\beta+1)a_2-N\Big\}c(x)|u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x\\
&\quad -\int_{\Omega}\langle x,\nabla c\rangle|u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x \\
& \geq {(\alpha+1)} \frac{N-p^+}{Np^+} \int_{\Omega}
|\nabla u|^{p(x)}{\mathrm{d}}x +{(\beta+1)}{\frac{N-q^+}{Nq^+}}
\int_{\Omega} |\nabla v|^{q(x)}{\mathrm{d}}x \\
&\quad - {(\alpha+1)} \frac{a_1}N\int_{\Omega}
|\nabla u|^{p(x)}{\mathrm{d}}x- {(\beta+1)}\frac{a_2}N
\int_{\Omega} |\nabla v|^{q(x)}{\mathrm{d}}x \\
&\quad + \int_{\Omega}\big\{(\alpha+1)a_1+(\beta+1)a_2-N\big\}c(x)
 |u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x\\
&\quad -\int_{\Omega}\langle x,\nabla c\rangle|u|^{\alpha+1}|v|^{\beta+1}{\mathrm{d}}x \\
&\geq \{(\alpha+1) \frac{N-p^+}{Np^+}  +{(\beta+1)}{\frac{N-q^+}{Nq^+}}
 - {(\alpha+1)} \frac{a_1}N\\
&\quad -{(\beta+1)}\frac{a_2}N\} 
 \int_\Omega c(x)|u|^{\alpha+1}|v|^{\beta+1}dx\\
&\geq \{(\alpha+1) \frac{N-p^+}{Np^+}  +{(\beta+1)}{\frac{N-q^+}{Nq+}}- 1\}
\int_\Omega c(x) |u|^{\alpha+1}|v|^{\beta+1}dx\\
&\quad  -\int_\Omega\langle x,\nabla c\rangle |u|^{\alpha+1}|v|^{\beta+1}dx.
\end{align*}

Now we remark that any solution $(u,v)$ of \eqref{2} satisfies 
$$
\int_\Omega c(x)|u|^{\alpha+1}|v|^{\beta+1}dx
=\int_{\Omega}|\nabla u|^{p(x)}{\mathrm{d}}x=\int_{\Omega}
|\nabla v|^{q(x)}{\mathrm{d}}x.
$$ 
So  from the hypothesis  \eqref{gradc}, the right-hand side is positive. 
A contradiction occurs, then the proof is complete.
\end{proof}


\section{Existence results via the fibering method}

Throughout this section, $\Omega$ denotes a bounded open set in 
$ \mathbb{R}^{N}$. The generalized Sobolev spaces $W_{0}^{1,p(x)}(\Omega)$ 
and $W_{0}^{1,q(x)}(\Omega)$ are equipped with the Luxembourg norm 
$\| u\|_{W_{0}^{1,p(x)}(\Omega)}$  and $\|u\|_{W_{0}^{1,q(x)}(\Omega)}$  
respectively. For a best reading, we denote as 
$\| u\|_{W_{0}^{1,p(x)}(\Omega)}=\| u|\|_{1,p(x)}$ and 
 $\| u\|_{W_{0}^{1,q(x)}(\Omega)}=\|u\|_{1,q(x)}$. 
Before starting this section, we need to make some crucial remarks 
for the understanding of this article. 

\begin{remark}\label{R1} \rm
Assuming that 
\[
(\alpha+1)\frac{N-p^-}{Np_-}+(\beta+1)\frac{N-q^-}{Nq^-}\leq1.
\]
We  can  establish that the term $\int_\Omega c(x)|
z|^{\alpha+1}| w|^{\beta+1}dx$ is well defined.
Indeed, since the functional  $c$ is bounded in $\Omega$, it suffices to verify 
that  $|z|^{\alpha+1}|w|^{\beta+1}$ belongs in $L^1(\Omega)$. 
This fact derives to the condition $\frac{{\alpha}+1}{p^+}+\frac{ \beta+1}{q^+}> 1$ 
and so $\frac{{\alpha}+1}{p^-}+\frac{ \beta+1}{q^-}> 1$. So, there exists a
pair $({\hat p},{\hat q})$ such that (1)
\begin{gather}  \label{pmoins}
{\ p}^-<{\hat p}<\frac{Np^-}{N-p^-}, \\
 \label{qmoins}
{\ q}^-<{\hat q}<\frac{Nq^-}{N-q^-}
\end{gather}
 (2)
$\frac{{\ \alpha}+1}{{\hat p}}+\frac{ \beta+1}{{\hat q}}= 1$.
\end{remark}

\begin{remark}\label{R2} \rm % \label{compact} 
Since $\frac{Np^-}{N-p^-}<\frac{Np(x)}{N-p(x)}$ and 
$\frac{Nq^-}{N-q^-}<\frac{Nq(x)}{N-q(x)}$, the assumption 
$(\alpha+1)\frac{N-p^-}{Np^-}+(\beta+1)\frac{N-q^-}{Nq^-}\leq1$ implies that
for any $x\in\Omega$, inequalities \eqref{pmoins} and \eqref{qmoins} become
\begin{gather*}
{\ p}^{-}<{\hat p}<\frac{Np(x)}{N-p(x)},\\
{\ q}{-}<{\hat q}<\frac{Nq(x)}{N-q(x)}.
\end{gather*}
In particular,  the imbeddings
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{{\hat p}}(\Omega)$ and 
$W_0^{1,q(x)}(\Omega)\hookrightarrow L^{{\hat q}}(\Omega)$ are continuous. 
Consequently, employing the H\"{o}lder inequality, the above estimate is fulfilled:
$$
\Big|\int_\Omega c(x)|u|^{\alpha+1}|v|^{\beta+1}dx\Big|
\leq \|c\|_{L^{\infty}(\Omega)}\|u\|_{L^{\hat{p}}(\Omega)}^{\alpha+1}
\|v\|_{L^{\hat{q}}(\Omega)}^{\beta+1}
\leq  C st\|u\|_{1,p(x)}^{\alpha+1}\|v\|_{1,q(x)}^{\beta+1}.
$$
\end{remark}

\begin{remark} \rm
(1) Under  assumption \eqref{alfa+}, we  have
\begin{equation}\label{alfa-}
\frac{\alpha+1}{p^{-}}+\frac{\beta+1}{q^{-}}-1>0.
\end{equation}

(2) When $q(x)$ and $p(x)$ are constant,  \eqref{alfa+} and \eqref{alfa-} are
 reduced to the well-known condition
\[
1<\frac{\alpha+1}p+\frac{\beta+1}q.
\]
\end{remark}

\subsection{Notation and hypotheses}

\subsubsection*{ Notation }

 $X_0(x)$ denotes $W_{0}^{1,p(x)}(\Omega) \times W_{0}^{1,q(x)}(\Omega)$.

 For any $(z,w)\in X_0(x)$, we set
\begin{equation}  \label{ABC} 
\begin{gathered}
A(z)=\int_{_{\Omega}}|\nabla z|^{p(x)} dx, \quad
B(w)=\int_{_{\Omega}}|\nabla w|^{q(x)} dx, \\
C(z,w)=\int_{\Omega}c(x)|z|^{\alpha+1}|w|^{\beta+1}dx.
\end{gathered}
\end{equation}

$$
\gamma^+= \frac{\alpha+1}{p^{+}}+\frac{\beta+1}{q^{+}},\quad 
\gamma^-= \frac{\alpha+1}{p^{-}}+\frac{\beta+1}{q^{-}}.
$$

 $\mathbf{J}$ designates the functional from  $X_0(x)$ to $\mathbb{R}$ and 
defined by
\begin{equation}\label{grandJ}
\mathbf{J}(u,v) = (\alpha+1)\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)} dx+
(\beta+1)\int_{\Omega}\frac{1}{q(x)}|\nabla v|^{q(x)}dx -C(u,v)dx.
\end{equation}

Following  remarks \ref{R1}-\ref{R2}, the functional $\mathbf{J}$  is  well defined 
from $X_0(x)$ to $\mathbb{I}$.

\subsubsection*{Hypotheses }
\begin{gather}\label{gamma} 
1<\gamma^+,\\
(p, q)\in \big({\mathcal P}(\Omega)\cup C(\overline{\Omega})\big)^2
\text{ satisfies \eqref{polder}.} \label{Hpq}
\end{gather}
 Moreover, assume that 
\begin{equation}\label{p-p+}
1<p^- \leq p^+<+\infty,\quad 1<q^- \leq q^+<+\infty. 
\end{equation}


\subsection*{Definition of a weak solution for \eqref{2}}

\begin{definition} \rm
A pair $(u,v)\in X_0(x)$ is a weak solution of \eqref{2}  if for any 
$(\phi,\psi) \in X_0(x)$:
\begin{gather*}
\int_{\Omega}| \nabla u|^{p(x)-2} \nabla u \nabla \phi\,dx
= \int _{\Omega}c(x)u |u|^{\alpha-1}|v|^{\beta+1} u \phi\,dx,   \\
\int_{\Omega}|\nabla v|^{q(x)-2}\nabla v \nabla \psi\,dx 
= \int _{\Omega}c(x)|u|^{\alpha+1} v|v|^{\beta-1} u \psi \,dx.   \\
\end{gather*}
\end{definition}

\subsection*{Fibering Method for quasilinear systems }
 Pohozaev  introduced the fibering method in \cite{stp} (see also \cite{sp,sipo}). 
For more details about various applications, we refer the reader to 
\cite{A,ah,abc,bm,bt,dp,i,km,as1,as2,tw,yw}).
The fibering method applied to this problem consists in seeking the
the pair $(u,v)\in X$ in the form
\begin{equation}\label{form}
u=rz,\quad v=\rho w
\end{equation}
where the functions $z$ and $w$ belong to $W^{1,p(x)}_{0}(\Omega)\setminus\{0\}$ and 
$W_{0}^{1,q(x)}(\Omega)\setminus\{0\}$ respectively,
where $r$ and $\rho$ are  real numbers. Moreover, since we look for nontrivial 
solutions (i.e: $u\neq 0$, $v\neq 0$),  we must assume that 
$r\neq0$ and $\rho \neq 0$. The fibering method ensures the existence. 
However, compared to other well known methods, we obtain the specific 
form \eqref{form}.

 \begin{remark}\label{t} \rm
In \cite{bm}, the authors  applied  the fibering method to obtain the existence 
of multiple solutions for a  problem like \eqref{2} when the 
exponents $p(x)$ and $q(x)$ are constant. In their studies, we note 
that the fibering parameters $r$ and $\rho$ depending on $z$ and $w$ verify 
 $r^p=\rho^q$ for $(z,w)$ such that $A(z)=1$ and $B(z)=1$ for instance.   
Inspired by this point of view, here we  propose  to seek a couple 
$(u,v)=(rz,\rho w)$, with $r=t^{1/p^+}$ and $\rho=t^{1/q^+}$, for $t>0$.
\end{remark}

\subsection*{Existence of a fibering parameter $t(z,w)$}
Existence and properties:
Since $\frac{\partial \mathbf{J}}{\partial u}(u,v)$ and 
$\frac{\partial \mathbf{J}}{\partial v}(u,v)$ exist,  a weak solution of 
\eqref{2} corresponds to a critical point of the energy functional $\mathbf{J}$ 
associated to the system \eqref{2}. 
Hence, assuming  that $(u,v) \in X_0(x) $ is a critical point of $\mathbf{J}$, 
 $(u,v)$ satisfies 
$\big(\frac{\partial \mathbf{J}}{\partial u}(u,v),
\frac{\partial \mathbf{J}}{\partial v}(u,v)\big)=(0,0)$. 
So, according to remark \ref{t}, a fibering
parameter $t(z,w)$ associated to $(z,w)$ is  characterized as 
\begin{equation}  \label{9}
\frac{d \mathbf{J}}{d t}\big(t^{1/p^+}z,t^{1/q^+}w\big)=0.
\end{equation}
More precisely, $t(z,w)$ is defined by the following Proposition.

\begin{proposition}\label{existrro} 
 Let $(z,w)$ be fixed in $X_0(x)$ such that $C(z,w)>0$.

(1) Assuming  \eqref{gamma}, there is $t(z,w)\in\mathbb{R}_+^*$
depending on $(z,w)$ such that
\begin{equation}  \label{zp}
\frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}t(z,w){^{\frac{p(x)}{p^+}}}|\nabla z|^{p(x)}dx
+\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}t(z,w){^{\frac{q(x)}{q^+}}}|\nabla z|^{q(x)}dx
={t(z,w)}^{{\gamma^+}}C(z,w).
\end{equation}

(2) Location of $t(z,w)$:
for $t(z,w)>1$ (respectively, $t(z,w)\leq 1$)  if $Q(z,w)>1$ 
(respectively $Q(z,w)\leq 1$),  for any $(z,w)$ such that 
$C(z,w)>0$, we have
$$
Q(z,w)=\frac{\frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}|\nabla z|^{p(x)}dx
+\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}|\nabla z|^{q(x)}dx}{C(z,w)}.
$$
 Moreover, the following two estimates hold:
(a) If $0<t(z,w)<1$, then
\begin{equation}\label{15}
Q(z,w)^{1/\gamma^+-1}\leq t(z,w)\leq Q(z,w)^{1/\gamma^+}
\end{equation}
(b) If $1\geq t(z,w)$, then
\begin{equation}  \label{16}
Q(z,w)^{1/\gamma^+}\leq t(z,w)\leq Q(z,w)^{1/\gamma^+-1}.
\end{equation}
\end{proposition}

\begin{proof}
 We divide the proof in three steps

\noindent{\bf Step 1: Existence of $t(z,w)$.}
Using the definition of $\mathbf{J}$ (see \eqref{grandJ}), solving \eqref{9} 
is equivalent to solving the equation
$$
\frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}t{^{\frac{p(x)}{p^+}}}|\nabla z|^{p(x)}dx
+\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}t{^{\frac{q(x)}{q^+}}}|\nabla z|^{q(x)}dx
=t^{{\gamma^+}}C(z,w).
$$
To do this, consider a function ${\tilde f}$ defined on $[0,+\infty[$ by
$$
{\tilde f}(t)= \frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}
t{^{\frac{p(x)}{p^+}}}|\nabla z|^{p(x)}dx+\frac{\beta+1}{q^+\gamma^+}
\int_{\Omega}t{^{\frac{q(x)}{q^+}}}|\nabla z|^{q(x)}dx-t^{{\gamma^+}}C(z,w).
$$
Choosing $0<t<1$, it follows that 
\begin{equation}\label{t1} 
t[Q(z,w)-t^{{\gamma^+-1}}]C(z,w) \leq{\tilde f}(t). 
\end{equation}
Now, for $1\leq t$, we obtain
\begin{equation}\label{1t}
{\tilde f}(t)\leq t[Q(z,w)-t^{{\gamma^+-1}}]C(z,w). 
\end{equation}
Consequently, on one side we have
$\lim_{t\to 0}{\tilde f}(t)\geq 0$, and on the other side
we have $\lim_{t\to+\infty}{\tilde f}(t)=-\infty$.
So, using the Mean Value Theorem, we deduce that there exists 
$t(z,w)\in\mathbb{R}_+^{*}$ depending on $z$, $w$ such that 
${\tilde f}(t(z,w))=0$. Moreover, $t(z,w)$ obeys to \eqref{zp}.

\noindent{\bf Step 2: Location of $t(z,w)$.}
We  distinguish the cases $Q(z,w)<1$ and $Q(z,w)\geq 1$.

(a) Assume that $Q(z,w)<1$, it follows that $0<t(z,w)<1$. Arguing  by opposite, 
if the assert  $1\leq t(z,w)$ holds, then from \eqref{1t}, we obtain 
$t(z,w)\leq Q(z,w)$. So, $Q(z,w)$ is greater than $1$. This is contradicts 
the hypothesis $Q(z,w)<1$.

(b) Conversely, assuming $Q(z,w)\geq 1$, 
from \eqref{t1},  we get $t(z,w)\geq 1$.

 From \eqref{zp} and the hypothesis \eqref{gamma}, it is easy to deduce that  
for any $(z,w)$  fixed in $X_0(x)$ such that $C(z,w)>0$, the location of
 the fibering parameter $t(z,w)$. 
\end{proof}

\begin{lemma}\label{HJ} 
Assume \eqref{gamma}.  Let $(z,w)$ in $X_{0}(x)\setminus\{(0,0)\}$ and $t(z,w)$ 
defined as in \eqref{zp}. The function
 $(z,w)\longmapsto t(z,w)$ is $C^1$ on $X_{0}(x)\setminus\{(0,0)\}$.
\end{lemma}

\begin{proof} 
 From \eqref{zp}, we consider on the open set 
$X_0(x)\setminus\{(0,0)\}\times(]0,1[\cup]1,+\infty[)$ of $X_0(x)\times\mathbb{I}$, 
the functional $\eta$ defined as follows:
$$
\eta(z,w,t)=\frac{\alpha+1}{p^+\gamma^+}\int_\Omega {t}^{\frac{p(x)}{p^+}
-\gamma^+}|\nabla z|^{p(x)}dx+\frac{\beta+1}{q^+\gamma^+}
\int_\Omega {t}^{\frac{q(x)}{q^+}-\gamma^+}|\nabla w|^{q(x)}dx-C(z,w).
$$
Obviously, we note that $\eta(z,w,t(z,w))=0$ and 
$\frac{\partial\eta}{\partial t}(z,w,t(z,w))< 0$.
We used the implicit function theorem for the function $\eta$. 
Then $(z,w)\longmapsto t(z,w)$ is $C^1$ function on 
$X_{0}(x)\setminus\{(0,0)\}$.
\end{proof}

\subsection*{A new definition for the enegy functional $\mathbf{J}$
derived from Proposition \ref{existrro} and  Lemma \ref{HJ}} 
On $X_0(x)\setminus\{(0,0)\}$, we  define  the function 
\begin{equation}\label{matcalJ}
\begin{aligned}
\mathcal{J}(z,w) &=\int_{\Omega}\frac{ \alpha+1}{p(x)}t(z,w)^{
\frac{p(x)}{p^+}}|\nabla z|^{p(x)}dx  +\int_{\Omega}\frac{ \beta+1}{q(x)
}t(z,w)^{ \frac{q(x)}{q^+}}|\nabla w|^{q(x)}dx\\ 
&\quad - {t(z,w)}^{\gamma^+}C(z,w).
\end{aligned}
\end{equation}

\subsection{A conditional critical point of $\mathcal{J}$}
We start by giving some lemmas.


\begin{lemma}\label{R}
 Let $(z_0,w_0)\in X_0(x)\setminus\{(0,0)\}$ such that $C(z_0,w_0)\neq 0$.
Then, there exists $Z_0\in W_0^{1,p(x)}(\Omega)\setminus\{0\}$ satisfying  
$C(Z_0,w_0)>0$.
\end{lemma}

\begin{proof} 
We fix $(z_0,w_0)\in X_0(x)\setminus\{(0,0)\}$ for which $C(z_0,w_0)\neq 0$.
Then distinguish two cases:
(1)  $C(z_0,w_0)>0$.  Then, the assertion of Lemma \ref{R} holds 
by taking $Z_0=z_0$.

(2) If $C(z_0,w_0)< 0$.  In this context, we note that
 $$
\int_\Omega c_+(x)|z_0|^{\alpha+1}|w_0|^{\beta+1}dx
< \int_\Omega c_-(x)|z_0|^{\alpha+1}|w_0|^{\beta+1}dx.
$$
Assuming that $c_+(x)> 0$ and $c_-(x)\geq 0$, we put   
$$
Z_0=z_0\,{\chi}_{_{\{h>0\}}}-{\hat\varepsilon}{z_0}\,{\chi}_{_{\{h\leq 0\}}}
$$
and
$$
0<{\hat\varepsilon}<\Big[\frac{\int_{\{c>0\}}h_+(x)|z_0|^{\alpha+1}
|w_0|^{\beta+1}dx}{\int_{\{c\leq 0\}}h_-(x)|z_0|^{\alpha+1}
 |w_1|^{\beta+1}dx+1} \Big]^{1/\alpha+1}.
$$
From easy calculations, it follows that
 $ \int_{\Omega}c(x)|Z_0|^{\alpha+1}|w_0|^{\beta+1}dx>0$.
 The proof  is complete.
\end{proof}

Consequently, we define the set  
\begin{equation}  \label{Econstraint}
E=\big\{(z,w)\in X;  \int_{\Omega}|\nabla z|^{p(x)}dx =1, \;
 \int_{\Omega}|\nabla w|^{q(x)}dx =1 \big\}.
\end{equation} 
 It is obvious that $E$  is a nonempty set (see \cite{xd, kr}).
We then have the next lemma.

\begin{lemma}\label{Eczw} 
The set $\{(z,w)\in E; C(z,w)>0 \}$ is nonempty. 
 \end{lemma}

\begin{proof}
 Let $(z_0,w_0)$ be in $X_0(x)\setminus\{(0,0)\}$ such that
 $C(z_0,w_0)\neq 0$. According to the lemma \ref{R}, there is 
$({\mathcal Z},w_0)\in X_0(x)\setminus\{(0,0)\}$  such that 
$C({\mathcal Z},w_0)>0$. The assert of the lemma is holds if for instance 
$({\mathcal Z},w_0)\in E$. Now, assume that $({\mathcal Z},w_0)$ is not in $E$. 
Assume that for instance $\int_\Omega|\nabla {\mathcal Z}|^{p(x)}dx>1$ and
$\int_\Omega|\nabla w_0|^{q(x)}dx<1$. Applying the mean value theorem
 to the functions $t\to 1-\int_\Omega |\nabla t{\mathcal Z}|^{p(x)}dx$
 and $s\to \int_\Omega |\nabla sw_0 |^{q(x)}dx -1$, we get a pair 
$(t_p,s_q)\in ]0,1[\times ]1,+\infty[$ such that 
$$
\int_\Omega |\nabla t_p{\mathcal Z}|^{p(x)}dx=1
=\int_\Omega |\nabla s_qw_0|^{q(x)}dx.
$$
Moreover, since $C({\mathcal Z},w_0)>0$, we also have 
$C(t_p{\mathcal Z},s_qw_0)>0$. 
The proof is complete.
\end{proof}

\begin{proposition}\label{caliJ} 
Let the functional $\mathcal{J}$ be defined by \eqref{matcalJ}, and let
$(z,w)$ be in  $E$. Under hypothesis \eqref{gamma}--\eqref{p-p+}, 
the following estimates hold:
\begin{gather*}
\frac{\gamma^+}{C(z,w)^{1/\gamma^+-1}}-1
\leq \mathcal{J}(z,w)\leq \frac{\gamma^--1}{C(z,w)
^{\min(\frac{p^-}{p^+},\frac{q^-}{q^+}) }}, \quad \text{if } c(z,w)\geq1,\\
\frac{\gamma^+-1}{C(z,w)^{\min(\frac{p^-}{p^+},\frac{q^-}{q^+})}}
\leq\mathcal{J}(z,w)\leq \frac{\gamma^-}{C(z,w)^{1/\gamma^+-1}}-1,\quad
 \text{if }c(z,w)<1.
\end{gather*}
\end{proposition}

\begin{proof} Estimates \eqref{15}  and  \eqref{16} imply  the
following lower and upper bounds for the functional $\mathcal{J}(z,w)$.
Indeed:
(1)  Consider $t(z,w)\geq 1$: after combining \eqref{matcalJ} and \eqref{zp}, it follows that
\begin{align*}
&\mathcal{J}(z,w)\\
&= (\alpha+1)\int_{\Omega}\big(\frac1{p(x)}-\frac1{p^+\gamma^+} \big)
 {t(z,w)}^{\frac{p(x)}{p^+}}|\nabla z|^{p(x)}dx\\
&\quad +(\beta+1)\int_{\Omega}\big(\frac1{q(x)}-\frac1{q^+\gamma^+} \big)
 {t(z,w)}^{\frac{q(x)}{q^+}}|\nabla w|^{q(x)}dx\\
&\geq \Big[ (\alpha+1)\frac{\gamma^+-1}{p^+\gamma^+}\int_{\Omega}|\nabla z|^{p(x)}dx 
\\ 
&\quad +(\beta+1)\frac{\gamma^+-1}{q^+\gamma^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]
 {t(z,w)}^{\min(\frac{p^-}{p^+},\frac{q^-}{q^+})}\\
&\geq \Big[ \frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}|\nabla z|^{p(x)}dx
 +\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]
(\gamma^+-1)Q(z,w)^{\min(\frac{p^-}{p^+},\frac{q^-}{q^+})}.
\end{align*}
The functional $\mathcal{J}(z,w)$ is bounded as follows:
\begin{align*}
\mathcal{J}(z,w)
& \leq t(z,w)\Big[\frac{\alpha+1}{p^-}\int_{\Omega}|\nabla z|^{p(x)}dx 
 +\frac{\beta+1}{q^-}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]-{t(z,w)}^{\gamma^+}C(z,w)\\
&\leq \Big[\frac{\alpha+1}{p^-}\int_{\Omega}|\nabla z|^{p(x)}dx 
 +\frac{\beta+1}{q^-}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]Q(z,w)^{1/\gamma^+-1}\\
&\quad - \Big[\frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}|\nabla z|^{p(x)}dx 
 +\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big].
\end{align*}

(2) Now, consider $t(z,w)< 1$:
\begin{align*}
&\mathcal{J}(z,w)\\
& \geq t(z,w)\Big[\frac{\alpha+1}{p^+}\int_{\Omega}|\nabla z|^{p(x)}dx 
 +\frac{\beta+1}{q^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]
 -{t(z,w)}^{\gamma^+}C(z,w)\\
&\geq \Big[\frac{\alpha+1}{p^+}\int_{\Omega}|\nabla z|^{p(x)}dx 
  +\frac{\beta+1}{q^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big]Q(z,w)^{1/\gamma^+-1}\\
&\quad- \Big[\frac{\alpha+1}{p^+\gamma^+}\int_{\Omega}|\nabla z|^{p(x)}dx 
+\frac{\beta+1}{q^+\gamma^+}\int_{\Omega}|\nabla w|^{q(x)}dx\Big].
\end{align*}
Also we have 
\begin{align*}
\mathcal{J}(z,w)
&=(\alpha+1)\int_{\Omega}\big(\frac1{p(x)}-\frac1{p^+\gamma^+} \big)
 {t(z,w)}^{\frac{p(x)}{p^+}}|\nabla z|^{p(x)}dx\\
& \quad +(\beta+1)\int_{\Omega}\big(\frac1{q(x)}-\frac1{q^+\gamma^+} \big)
 {t(z,w)}^{\frac{q(x)}{q^+}}|\nabla w|^{q(x)}dx\\
&\leq \Big[ (\alpha+1)\big(\frac1{p^-}-\frac1{p^+\gamma^+}\big)
 \int_{\Omega}|\nabla z|^{p(x)}dx\\ 
&\quad +(\beta+1)\big(\frac1{q^-}-\frac1{q^+\gamma^+}\big)
 \int_{\Omega}|\nabla w|^{q(x)}dx\Big]
 {t(z,w)}^{\min(\frac{p^-}{p^+},\frac{q^-}{q^+})}\\
&\leq \Big[(\alpha+1)\big(\frac1{p^-}-\frac1{p^+\gamma^+}\big)
 \int_{\Omega}|\nabla z|^{p(x)}dx \\
&\quad +(\beta+1)\big(\frac1{q^-}-\frac1{q^+\gamma^+}\big)
\int_{\Omega}|\nabla w|^{q(x)}dx \Big]Q(z,w)^{\min
(\frac{p^-}{p^+\gamma^+},\frac{q^-}{q^+\gamma^+})}.
\end{align*}
We choose $(z,w)\in E$, then $Q(z,w)$ is reduced to become 
$Q(z,w)=\frac1{C(z,w)}$. Thus, the assert of the Proposition \ref{caliJ} follows.
\end{proof}

Consider  the optimal problem \begin{equation}\label{inf}
\inf_{\{(z,w)\in E; \,\, c(z,w)>0\}} \frac{1}{C(z,w)}.
\end{equation}
We claim that the infimum  value is attained  in $E$.
  To assert this claim, we need the following lemma.


\begin{lemma}\label{supC}
 Under  assumption \eqref{Hpq},
the optimal problem \eqref{inf} possesses  at least one solution.
\end{lemma}

\begin{proof}
Solving \eqref{inf}  is equivalent to solving the maximizing problem:
\begin{equation}\label{M}
\sup \Big\{\int_{\Omega}c(x)| z|^{\alpha+1}| w|^{\beta+1}dx; \,
 (z,w)\in E,\; C(z,w)>0\Big\} =:M.
\end{equation} 
Firstly, from Remarks \ref{R1} and \ref{R2}, we observe that $M$ is finite. 
Indeed, from the end of Remark \ref{R2} and the use of \cite{xd} or \cite{kr}, 
for any $(z,w)\in E$, $0<C(z,w)\leq \|c\|_{\infty}K$, (the constants 
$\|c\|_{\infty}$ and $K$ are not depending on $(z,w)$).
We follow the ideas of \cite{bm} and we show
that there exists $(z_{M},w_{M})\in E$ such that 
$C(z,w)\leq C(z_{M},w_{M})$ for any $(z,w)\in E$. 

Let $(z_{n},w_{n})$ be a maximizing sequence of \eqref{M} (i.e $(z_n,w_n)$ 
is such that
$ A(z_{n})=1,\quad  B(w_{n})=1$ and  $C(z_n,w_n) \to M >0)$.
It is easy to see that  $(z_{n},w_{n})$ is bounded in $ X_{0}(x)$.
It follows that
$ z_{n}\rightharpoonup \overline{z}$ weakly in  $ W_{0}^{1.p(x)}(\Omega) $ 
and $z_{n}\to \overline{z}$ strongly in   $L^{\hat{p}}(\Omega)$.
Similarly,  $w_{n}\rightharpoonup\overline{w}$ weakly in   
$ W_{0}^{1.q(x)}(\Omega) $
 and $z_{n}\to \overline{z}$  strongly  in $L^{\hat{q}}(\Omega)$.
Consequently
$$ 
C(z_n,w_n)\to C(\bar{z},\bar{w}).
$$
Moreover, since $ z\mapsto \int_\Omega |\nabla z|^{p(x)}dx$ is a semimodular 
in the sense of \cite[Definition 2.1.1]{dhhr},  
applying  \cite[Theorem 2.2.8]{dhhr}), 
we obtain that $p(x)$- and $q(x)$-modular functions 
$\rho_p(\cdot)$  and $\rho_q(\cdot)$ are
weakly lower semicontinuous.  
So, since  $z_{n}\rightharpoonup \bar{z}$ 
weakly in  $ W_{0}^{1.p(x)}(\Omega)$, we deduce that
$$
\int_{\Omega}| \nabla\bar{ z}|^{p(x)}\,dx
\leq\liminf_{n} \int_{\Omega}| \nabla \bar{ z_{n}}|^{p(x)}\,dx=1
$$
and 
$$
\int_{\Omega}| \nabla\bar{ w}|^{q(x)}\,dx
\leq\liminf_{n}\int_{\Omega} | \nabla \bar{ w_{n}}|^{q(x)}\,dx=1.
$$
Now assume by contradiction  that
$ \int_{\Omega}| \nabla \bar{ z}|^{p(x)}\,dx<1$ and 
$ \int_{\Omega}| \nabla \bar{w}|^{p(x)}\,dx<1$. Then we  have 
$\|\bar{z}\|_{1,p(x)}<1$  and $\|\bar{w}\|_{1,p(x)}<1$.
We set
$$ 
a=\|\bar{z}\|_{1,p(x)}= \|\nabla \bar{z} \|_{L^{p(x)}},\quad
b = \|\bar{w}\|_{1,q(x)}=\|\nabla \bar{w} \|_{L^{q(x)}}.
$$
Using again the properties of the functions $\rho_p$ and
$\rho_q$, it follows that 
$$
\rho_p\big(|\nabla(\frac{1}{a}\bar{ z})|\big)
=\int_{\Omega}|\nabla(\frac{1}{a} \bar{ z})|^{p(x)}\,dx=1
$$
 and 
$$
\rho_q\big(|\nabla (\frac{1}{b}\bar{ w})|\big)
=\int_{\Omega}|\nabla(\frac{1}{b} \bar{ w})|^{q(x)}\,dx=1.
$$  
Obviously, we see that
$\big(\frac{1}{a}\bar{z},\frac{1}{b}\bar{w}\big)\in E$.
On the other hand,
$$
C\big(\frac{1}{a}z_n,\frac{1}{b}w_n\big) \to  
C\big(\frac{1}{a}\bar{z},\frac{1}{b}\bar{w}\big) \quad \text{as } n\to +\infty.
$$
 However, we remark that
$$
C\big(\frac{1}{a}\bar{z},\frac{1}{b}\bar{w}\big)
=\big(\frac{1}{a}\big)^{\alpha+1}\big(\frac{1}{b}\big)^{\beta +1}C(\bar{z},\bar{w})
= \big(\frac{1}{a}\big)^{\alpha+1}\big(\frac{1}{b}\big)^{\beta +1}M.
$$
Since $a<1$ and $b<1$, we obtain 
$C\big(\frac{1}{a}\bar{z},\frac{1}{b}\bar{w}\big)>M$.
A contradiction occurs.
\end{proof}

Consequently, combining Proposition \ref{caliJ} and  Lemma \ref{supC},  
we  deduce that 
$$
\inf_{\{(z,w)\in E;\,\, C(z,w)>0\}}{\mathcal{J }}(z,w)
$$
 exists.
In the next section, we will show that the infimum of the functional 
${\mathcal{J }}(z,w)$ is attained on $E$.

\subsection*{Existence result for the optimal problem \eqref{inf}}
We are looking for  $(z,w)\in E $ satisfying
\begin{equation}\label{inftildeJ}
\inf {\mathcal{J}}(z,w): A(z)=1,\; B(w)=1.
\end{equation}
To investigate \eqref{inftildeJ}, we give some lemmas and remarks.

\begin{lemma}\label{tutetav}
Let $E$ be the set defined as in \eqref{Econstraint}. 
Assume that the functions $p$ and $q$ satisfy  hypothesis \eqref{Hpq}.
Then, for any $(z,w) \in X_{0}(x)$,  there exit $ \delta(z)>0$  and
$\theta(w)>0$ such that
\begin{equation*}
\big(\frac{1}{\delta(z)}z,\frac{1}{\theta(w)}w\big) \in E.
\end{equation*}
\end{lemma}

\begin{proof} 
For any fixed $z$ in $W^{1,p(x)}_{0}(\Omega)\setminus\{0\}$, we define a 
function $f$ on $]0,+\infty[$ by
 \begin{equation*}
f(z,\delta)= \int_\Omega (\frac{1}{\delta})^{p(x)}|\nabla z|^{p(x)}dx-1.
\end{equation*}
For any $\delta>1$, we have
\begin{equation*}
(\frac1\delta)^{p_+}\int_\Omega |\nabla
z|^{p(x)}dx-1\leq f(z,\delta)
\leq (\frac1\delta)^{p^-} \int_\Omega|\nabla z|^{p(x)}dx-1.
\end{equation*}
Now, taking $\delta<1$, we obtain
\begin{equation*}
(\frac1\delta)^{p^-}\int_\Omega|\nabla z|^{p(x)}dx-1
\leq f(z,\delta)\leq \big(\frac1\delta\big)^{p^+}
\int_\Omega |\nabla z|^{p(x)}dx-1.
\end{equation*}
It follows from the above inequality that
\begin{itemize}
\item for $\delta$ large enough, $f(z,\delta)\to -1$  as $\delta\to +\infty$,
\item  for $\delta$ small enough, $f(z,\delta)\to +\infty$  as $\delta\to 0$.
\end{itemize}
By applying the Mean Value Theorem, we conclude that there exists 
$\delta_z\in ]0,+\infty[$ such that 
\begin{equation*}
\int_\Omega \frac1{\delta_z^{p(x)}}|\nabla z|^{p(x)}dx=1.
\end{equation*}
Similarly, we can prove that, there exists $\theta_w>0$ such that:
\begin{equation*}
\int_\Omega \frac1{\theta_w^{q(x)}}|\nabla w|^{q(x)}dx=1.
\end{equation*}
The proof is complete.
\end{proof}

\begin{lemma} Let $(z,w) \in X_{0}(x)$ be fixed.
The functions $z\mapsto \delta(z)$ defined in Lemma \ref{tutetav} possess 
$C^{1}$-regularity respectively from $\mathcal{U}_{z,\delta_z}$ to
 $\mathbb{I}$ and $\mathcal{V}_{w,\theta_w}$ to $\mathbb{I}$.
Here,   $\mathcal{U}_{z,\delta_z}$  is a neighborhood  of 
$({z,\delta_z})$ lying on the open set  
$\mathcal{U}={W}_0^{1,p(x)}(\Omega)\setminus\{0\}\times ]0,+\infty[$ and 
$\mathcal{V}_{w,\theta_w}$ is a neighborhood  of $(w,\theta_w)$ lying on the 
open set $\mathcal{V}=W_0^{1,q(x)}(\Omega)\setminus\{0\}\times ]0,+\infty[$.
\end{lemma}

\begin{proof}
After making a simple computation,  it is easily to see that
\begin{equation*}
\frac{\partial f}{\partial \delta}(z,\delta)=-\frac1\delta\int_\Omega
p(x)\frac1{\delta^{p(x)}}|\nabla z|^{p(x)}dx.
\end{equation*}
Replacing $\delta$ by $\delta_{z}$, we have
\begin{equation*}
| \frac{\partial f}{\partial \delta}(z,\delta_z)| >\frac{p^-}{\delta_z}>0.
\end{equation*}
Hence,  the implicit function theorem implies that there exist a neighborhood 
of $({z,\delta_z})$, $\mathcal{U}_{z,\delta_z}\subset\mathcal{U}$
and a function of class $C^{1}:$ $z\longmapsto \delta(z)$ from
$\mathcal{U}_{z,\delta_{z}}$  to  $\mathbb{I}$.
Particularly, for all  $z$  in  $ W_{0}^{1,q(x)}(\Omega)$, we have
\begin{equation}\label{n1}
\delta'(z)\cdot \phi=- \frac{\frac{\partial f}{\partial z}
(z,\delta_{z})\cdot\phi}{\frac{\partial f}{\partial \delta}(z,\delta_{z}) }.
\end{equation}
Since we have
\begin{equation*}
\frac{\partial f}{\partial z}(z,\delta_z)\cdot\phi= \int_\Omega
p(x)\frac1{\delta_z^{p(x)}}|\nabla z|^{p(x)-2}\nabla z\cdot\nabla \phi dx,
\end{equation*}
 the definition \eqref{n1} then becomes
\begin{equation}  \label{tprimez}
\delta'(z)\cdot\phi=-\frac{\int_\Omega
p(x)\frac1{\delta_z^{p(x)}}|\nabla z|^{p(x)-2}\nabla z\cdot\nabla \phi dx}{
\frac1\delta_z\int_\Omega p(x)\frac1{\delta_z^{p(x)}}|\nabla z|^{p(x)}dx}.
\end{equation}
In the same way, we  have
\begin{equation}  \label{thetaprimew}
\theta'(w)\cdot\psi=-\frac{\int_\Omega
q(x)\frac1{\theta_w^{q(x)}}|\nabla w|^{q(x)-2}\nabla w\cdot\nabla \psi dx}{
\frac1\theta_w\int_\Omega q(x)\frac1{\theta_w^{q(x)}}|\nabla
w|^{q(x)}dx.}.
\end{equation}
\end{proof}


\begin{remark} \rm
We introduce the functional ${\bf{\widetilde {J}}}$ defined on 
${W_0^{1,p(x)}}\times{W_0^{1,q(x)}}\times\mathbb{I}$ by
 \begin{equation}\label{barJ}
 {\bf{\widetilde {J}}}(z,w,t)={\bf{{J}}}(t^{1/p^+}z,t^{1/q^+}{w}).
 \end{equation}
Thus,  for any $(z,w)\in X_0(x)\setminus\{(0,0)\}$ and $t(z,w)$ given by \eqref{zp}, 
this definition implies that
\begin{equation}
 {\bf{\widetilde {J}}}(z,w,t(z,w))={\mathcal{J}}(z,w)
\end{equation}
where the functional ${\mathcal{J}}$ is given by \eqref{matcalJ}.
\end{remark}

\begin{lemma}\label{minimizing}
Let $(z_{n},w_{n}) \in E $ be a minimizing sequence of \eqref{inftildeJ}, 
the sequence $(u_n,v_n)$ with
$$
u_{n}=t(z_{n},w_{n})^{1/p^+}z_{n},\quad 
v_{n}=t(z_{n},w_{n})^{1/q^+}w_{n}
$$
is then a Palais-Smale sequence for the functional $\mathbf{J}$. i.e.,
 %\label{A:gp}
\begin{gather}
\mathbf{J}(u_n,v_n)\leq m,\label{1:gp1}\\
\mathbf{J}'(u_n,v_n)\to 0,  \text{ in the meaning of the norm } 
\|\cdot\|_{X_0^{*}(x)}.\label{A: gp2}
\end{gather}
\end{lemma}

\begin{proof} 
We  follow  the ideas of \cite{ah}. For a best understanding, some of the notation 
used here remain unchanged.
Generalizing \cite{ah}, we define  
$\pi:W_0^{1,p(x)}(\Omega)\setminus\{0\} \to \mathbb{I}$ by
$$
\pi(z)= (\pi_1(z),\pi_2(z))=\big(\delta(z), \frac{z}{\delta(z)}\big)
$$ 
and  $\tau: W_0^{1,q(x)}(\Omega)\setminus\{0\} \to \mathbb{I}$ by 
$$
\tau(w)= (\tau_1(w),\tau_2(w))=\big(\theta(w), \frac{w}{\theta(w)}\big).
$$
Before continuing, let us designate by $T_{(z,w)}E$ the tangent space to $E$. 
Denote 
$$
E_p=\{z\in W_0^{1,p(x)}(\Omega); A(z)=1\}
$$ 
(respectively, $E_q=\{w\in W_0^{1,q(x)}(\Omega); B(z)=1\}$),
 hence, it is clear that $T_{(z,w)}E=T_zE_p\times T_wE_q$.
Moreover, for any $(z,w)\in X_0(x)$, for any $(\Phi,\Psi)\in T_{(z,w)}E$,
 we have
$$
{\mathcal{J}}'(z,w)(\Phi,\Psi)=\frac{\partial{\bf\widetilde{J}}}{\partial z}
(z,w,t(z,w))(\Phi)+\frac{\partial{\bf\widetilde{J}}}{\partial w}(z,w,t(z,w))(\Psi). 
$$
 Now, we consider a minimizing sequence $(z_{n},w_{n})\in E$. For any 
$(\phi,\psi)\in X_0(x)$, it is obvious that 
$(\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi)\in T_{(z,w)}E$.

 From the above, setting $B_n=(z_n,w_n,t(z_n,w_n))$ and following the spirit 
of the proof of the \cite[Lemma 3.1]{ah},  we have:
  \begin{gather*}
  \frac{\partial{\bf{J}}}{\partial u}(u_n,v_n)(\phi)
 = \frac{\partial{\bf\widetilde{J}}}{\partial z}(B_n)(\pi'_2(z_n)\cdot\phi),\\
   \frac{\partial{\bf{J}}}{\partial v}(u_n,v_n)(\phi)
 = \frac{\partial{\bf\widetilde{J}}}{\partial w}(B_n)(\pi'_2(w_n)\cdot\psi),\\
  {\mathcal{J}}'(z_n,w_n)(\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi)
=\frac{\partial{\bf\widetilde{J}}}{\partial z}(B_n)(\pi'_2(z_n)\cdot\phi)
 +\frac{\partial{\bf\widetilde{J}}}{\partial w}(B_n)(\tau'_2(w_n)\cdot\psi). 
 \end{gather*}
Then, since 
$$
\mathbf{J}'(u_n,v_n)(\phi,\psi)
=\frac{\partial{\bf{J}}}{\partial u}(u_n,v_n)(\phi)
+\frac{\partial{\bf{J}}}{\partial v}(u_n,v_n)(\psi)
$$ 
for any $(\phi,\psi)\in X_0(x)$,
it follows that
$$
{\bf{J}}'(u_n,v_n)(\phi,\psi)={\mathcal{J}}'(z_n,w_n)
(\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi). 
$$
However, applying the Ekeland  variational principle, we have
$$
|\mathcal{J}{'}(z_n,w_n)(\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi)|
\leq\frac{1}{n}\| (\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi)\|_{X_{0}(x)},
$$
for all $(\phi,\psi)\in X_0(x)$.
Therefore, 
\begin{equation*}
| {\bf {J}}'(u_n,v_n)\cdot(\phi,\psi)| 
\leq \frac1n \|\big(\pi'_2(z_n)\cdot\phi,\tau'_2(w_n)\cdot\psi\big)\|_{X_0(x)},
\quad \forall (\phi,\psi)\in X_0(x).
\end{equation*}
The space $X_0(x)$ is equipped with the cartesian norm 
$\|\cdot \|_{X_0(x)}=\|\cdot \|_{1,p(x)}+\|\cdot \|_{1,q(x)}$.
Then the following estimate holds
\begin{equation}\label{Jpi2}
| {\bf {J}}'(u_n,v_n)\cdot(\phi,\psi)|  \leq
\frac1n\Big(\|(\pi'_2(z_n)\cdot\phi\|_{1,p(x)} 
+ \| \tau'_2(w_n)\cdot\psi)\|_{1, q(x)}\Big).
\end{equation}
To simplify notation, we set ${\tilde\delta}_n=\delta(z_n)$.
 So, from the definition of $\pi_2$, we check that
\begin{equation*}
\pi_2'(z_n) \cdot\phi=\frac{\phi}{{\tilde\delta}_n
}-\frac{z_n\int_\Omega p(x)\frac1{{\tilde\delta}
_n^{p(x)}}|\nabla z_n|^{p(x)-2}\nabla z_n\cdot\nabla \phi
dx}{\frac1{\tilde\delta}_n\int_\Omega p(x)\frac1{{\tilde\delta}
_n^{p(x)}}|\nabla z_n|^{p(x)}dx}.
\end{equation*}
Thus,
\begin{align*}
\|\pi_2'(z_n) \cdot\phi\|_{1,p(x)}
& \leq  \frac{\|\phi\|_{1,p(x)}}{{\tilde\delta}_n}+\frac{\|z
_n\|_{1,p(x)}|\int_\Omega p(x)\frac1{{\tilde\delta}
_n^{p(x)}}|\nabla z_n|^{p(x)-2}\nabla z_n\cdot\nabla \phi
dx|}{\frac1{\tilde\delta}_n\int_\Omega p(x)\frac1{{\tilde\delta}
_n^{p(x)}}|\nabla z_n|^{p(x)}dx} 
 \\
& \leq  \frac{\|\phi\|_{1,p(x)}}{{\tilde\delta}_n}+\frac{|
\int_\Omega p(x)\frac1{{\tilde\delta}_n^{p(x)}}|\nabla z
_n|^{p(x)-2}\nabla z_n\cdot\nabla \phi dx|}{
\int_\Omega p(x)\frac1{{\tilde\delta}_n^{p(x)}}|\nabla z_n|^{p(x)}dx}.
\end{align*}
Particularly, applying successively the H\"{o}lder inequality for  
$p(x)$-Lebesgue space \cite{kwz,kr,xd}, we find
% \label{B:gp}
\begin{gather}
\begin{aligned}
\big|\int_\Omega p(x) \frac{|\nabla z_n|^{p(x)-2}}{z_n^{p(x)-2}}
\frac{\nabla z_n}{{\tilde\delta}_n}\cdot\frac{
\nabla \phi}{{\tilde\delta}_n}dx\big| 
& \leq p+\big|\frac{|\nabla z
_n|^{p(x)-1}}{{\tilde\delta}_n^{p(x)-1}} \big|_{L^{\frac{p(x)}{p(x)-1}
}(\Omega)} \frac{\|\phi\|_{1,p(x)}}{{\tilde\delta}_n} \\
& =p^+ \frac{\|\phi\|_{1,p(x)}}{{\tilde\delta}_n}.
\end{aligned}\label{B:gp1}
\\
\int_\Omega p(x)\frac1{{\tilde\delta}_n^{p(x)}}|\nabla
z_n|^{p(x)}dx\geq p^{-}\int_\Omega \frac1{{\tilde\delta}
_n^{p(x)}}|\nabla z_n|^{p(x)}dx\geq p^-.\label{B;gp2}
\end{gather}
The above remarks allow us to obtain the new estimate:
\begin{equation*}
\|\pi_2'(z_n) \cdot\phi \|_{1,p(x)}
\leq \big(1+\frac{p+}{p^-} \big) \frac{\|\phi\|_{1,p(x)}}{{\tilde\delta}_n}.
\end{equation*}
From the properties on the spaces 
$L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$ spaces 
(see for instance \cite{xd}), and because  
$\int_\Omega\frac{|\nabla z_n|^{p(x)}}{{\tilde\delta}_n^{p(x)}}dx=1$ 
and $\int_\Omega |\nabla z_n|^{p(x)}dx=1$, we have $\|z_n\|_{1,p(x)}={\tilde\delta}_n=1$.
 Therefore
\begin{equation*}
\|\pi_2'(z_n) \cdot\phi\|_{1,p(x)}
\leq\big(1+\frac{p^+}{p^-} \big) \|\phi\|_{1,p(x)}.
\end{equation*}
Similarly,
\begin{equation*}
\|\tau_2'(w_n) \cdot\psi \|_{1,q(x)}
\leq \big(1+ \frac{q^+}{q^-} \big)\|\psi\|_{1,q(x)}.
\end{equation*}
Taking into account the estimate \eqref{Jpi2}, we conclude that
\begin{equation*}
\lim_{n\to+\infty}\|\mathbf{J}'(u_n,v_n) \|_{X_0^*(x)}=0.
\end{equation*}
This completes the proof.
\end{proof}

\begin{lemma}\label{L}
Assume that \eqref{gamma} holds.  
Let $(z_n,w_n)$ be a minimizing sequence of $\mathbf{J}$ on the manifold $E$.
The sequence $(u_n,v_n)=(t(z_n,w_n)^{1/p^+}z_n,t(z_n,w_n)^{1/q^+}w_n)$ is bounded 
in  $ X_0(x)$.
\end{lemma}

\begin{proof}
Since $u_n=t(z_n,w_n)^{1/p^+}z_n$, $v_n=t(z_n,w_n)^{1/q^+}w_n$, by
 the characterization \eqref{zp}, it follows that
\begin{equation}  \label{Jprimen}
\int_\Omega |\nabla u_n|^{p(x)}dx
+ \int_\Omega |\nabla v_n|^{q(x)}dx
 -2\int_\Omega c(x)| u_n|^{\alpha+1}| v_n|^{\beta+1}dx=0.
\end{equation}
On the other hand, because $(z_n,w_n)$ is a minimizing sequence for 
$\inf_{(z,w)\in E}{\mathcal {J}}(z,w)$, we have
\begin{equation}  \label{m}
m\leq (\alpha+1)\int_\Omega \frac1{p(x)} |\nabla
u_n|^{p(x)}dx+(\beta+1)\int_\Omega \frac1{q(x)} 
|\nabla v_n|^{q(x)}dx-C(u_n,v_n)<m+\frac1n.
\end{equation}
Combining \eqref{Jprimen} and \eqref{m}, one concludes that
\begin{equation*}
m\leq \int_\Omega \big( \frac{\alpha+1}{p(x)} -1\big)
|\nabla u_n|^{p(x)}dx+\int_\Omega \big( \frac{\beta+1
}{q(x)} -1\big) |\nabla v_n|^{q(x)}dx+C(u_n,v_n)<m+\frac1n.
\end{equation*}
Recall that
\begin{equation*}
\int_\Omega |\nabla u_n|^{p(x)}dx=
\int_\Omega |\nabla v_n|^{q(x)}dx=\int_\Omega c(x)| u_n|^{\alpha+1}| v_n|^{\beta+1}dx,
\end{equation*}
hence
\begin{equation*}
m\leq \int_\Omega \frac{\alpha+1}{p(x)} |\nabla
u_n|^{p(x)}dx+\int_\Omega \big( \frac{\beta+1}{q(x)} -1\big)
|\nabla v_n|^{q(x)}dx <m+\frac1n.
\end{equation*}
After making some easy calculations, we obtain
\begin{equation*}
\int_\Omega |\nabla u_n|^{p(x)}dx<\frac{m+1}{\gamma^+-1}.
\end{equation*}
Arguing similarly,  we find that
\begin{equation*}
\int_\Omega |\nabla v_n|^{q(x)}dx<\frac{m+1}{\gamma^+-1}.
\end{equation*}
We have proved that the sequence is bounded in $X_{0}(x)$. 
\end{proof}


\begin{lemma}\label{wcv} 
Under  hypothesis \eqref{Hpq},  problem \eqref{inftildeJ} possesses  
at least one solution.
\end{lemma}

\begin{proof}    We divide the proof in three steps.

\noindent\textbf{Step 1:} Weak convergence of $u_n$ and $v_n$.
Let $(z_{n},w_{n})\in E $ be a minimizing  sequence.
From Lemma \ref{minimizing}, it is known  that
$\lim_{n\to+\infty}\mathbf{J}(u_{n},v_{n})=m $ and $
\lim_{n\to+\infty}\|\mathbf{J}'(u_n,v_n)
\|_{X_0^*(x)}=0 $ and that  $(u_n,v_n)$is bounded in $X_{0}(x)$. 
Extracting if necessary to a subsequence,  there exists a pair $({u^*},{v^*})$  
in $X_{0}(x)$ such that
\begin{gather*}
u_n\rightharpoonup {u^*}\quad\text{in }W_{0}^{1.p(x)}(\Omega),\\
v_n\rightharpoonup {v^*}\quad\text{in }W_{0}^{1.q(x)}(\Omega).
\end{gather*}

\noindent\textbf{Step 2:} Strong convergence of $u_n$ and $v_n$ in 
$W_0^{1,p(x)}(\Omega)$ (resp. $W_0^{1,q(x)}(\Omega)$).
To do this, we establish that $u_n$ and $v_n$ are two Cauchy sequences.
Firstly, easy calculations ensure that for any $m\in\mathbb{N}$ and $l\in\mathbb{N}$,
\begin{align*}
&\big[ \mathbf{J}'(u_{m},v_{m})-\mathbf{J}'(u_{l},v_{l})
\big](u_{m}-u_{l},0)\\
&= (\alpha+1)\int_{\Omega}(|\nabla u_{m}|^{p(x)-2}\nabla u_{m}
-|\nabla u_{l}|^{p(x)-2}\nabla u_{l})(\nabla u_{m}-u_{l})dx\\
&\quad - (\alpha+1)\int_{\Omega}c(x)\big[| v_{m}|^{(\beta+1}
 | u_{m}| ^{\alpha-1} u_{m}-
| v_{l}|^{(\beta+1}| u_{l}|
^{\alpha-1}u_{l}\big](u_{m}-u_{l})dx.
 \end{align*}
 Thus, after making a suitable rearrangement, we obtain
 \begin{align*}
&\int_{\Omega}(|\nabla u_{m}|^{p(x)-2}\nabla u_{m}
-|\nabla u_{l}|^{p(x)-2}\nabla u_{l})(\nabla u_{m}-u_{l})dx\\
&= \frac{1}{\alpha+1}\big[ \mathbf{J}'(u_{m},v_{m})-\mathbf{J}'(u_{l},v_{l})
\big](u_{m}-u_{l},0)dx\\
&\quad + \int_{\Omega}c(x)\big[| v_{m}|^{\beta+1}| u_{m}|
^{\alpha-1}u_{m}-
| v_{l}|^{\beta+1}| u_{l}|
^{\alpha-1}u_{l}\big](u_{m}-u_{l})dx.
 \end{align*}
We claim that
 \begin{equation}\label{ml}
 \int_{\Omega}c(x)\big[| v_{m}|^{(\beta+1}| u_{m}|
^{\alpha-1}u_{m}-
| v_{l}|^{(\beta+1}| u_{l}|
^{\alpha-1}u_{l}\big](u_{m}-u_{l})dx\to 0, 
\end{equation}
as $m,l\to +\infty$.  
Indeed in view of \eqref{ml} and according to Remarks \ref{R1} and \ref{R2} 
(the notation used remains the same), we observe that
\begin{equation}\label{uml}
\begin{aligned}
&\big|\int_{\Omega}c(x)\big[| v_{m}|^{\beta+1}| u_{m}|
^{\alpha-1}v_{m}-
| v_{l}|^{\beta+1}| u_{l}|
^{\alpha-1}u_{l}\big](u_{m}-u_{l})dx\big|\\
&\leq \int_{\Omega}c(x)| v_{m}|^{\beta+1} | u_{m}|
^{\alpha}| u_{m}-u_{l} | dx+\int_{\Omega}c(x)
| v_{l}|^{\beta+1} | u_{l}|
^{\alpha}| u_{m}-u_{l} | dx\\
& \leq \|c\|_{\infty} \| v_{m} \|_{{L}^{\hat{q}}(\Omega)}  ^{\beta+1}
\| u_{m} \|_{{L}^{\hat{p}}(\Omega)}^\alpha
\| u_{m}-u_{l}  \|_{{L}^{\hat{p}}(\Omega)}\\
&\quad +\|c\|_{\infty}\| v_{l} \|_{{L}^{\hat{q}}(\Omega)}  ^{\beta+1}
\| u_{l} \|_{{L}^{\hat{p}}(\Omega)}^\alpha
\| u_{m}-u_{l}  \|_{{L}^{\hat{p}}(\Omega)}\\
&\leq  C \| u_{m}-u_{l}  \|_{{L}^{\hat{p}}(\Omega)}.
\end{aligned}
\end{equation}
Before continuing, we recall a fundamental convergence property.
It is well known that the imbedding 
$W_{0}^{1.p(x)}(\Omega)\hookrightarrow L^{\delta(x)}(\Omega)$ 
(resp $W_{0}^{1.q(x)}(\Omega)\hookrightarrow L^{\gamma(x)}(\Omega)$) 
 with $\delta(x)<\frac{Np(x)}{N-p(x)}$ (resp. $\gamma(x)< \frac{Nq(x)}{N-q(x)}$)
 is compact  (see \cite{xd}).

Choose $\gamma(x)=\hat{p}$, it follows that $u_n$ converges strongly to 
$u^*$ in $L^{\hat p}(\Omega)$ and so on, $u_n$ is a Cauchy sequence in sense 
of the $L^{\hat p}(\Omega)$ norm. Consequently,
\eqref{uml} occurs.
Furthermore, following \cite{kwz}, there exist constants $C_1, C_2, C_3, C_4$ 
such that
\begin{equation}\label{F}
\langle F(\nabla u_m)-F(\nabla u_l), u_m-u_l \rangle
\geq \begin{cases}
 C_1\|u_m-u_l\|_{1,p(x)}^2,& \text{if } 1<p(x)<2,\\ 
 C_2\|u_m-u_l\|_{1,p(x)}^{2p_{0,1}/p_{1,1}}, & \text{if } 1<p(x)<2,\\
 C_3\|u_m-u_l\|_{1,p(x)} ^{p_{0,2}}, &\text{if } 2\geq p(x),\\ 
 C_4\|u_m-u_l\|_{1,p(x)}^{p_{1,2}},&\text{if } 2\geq p(x),
\end{cases}
\end{equation}
where
 $F(\xi)=|\xi|^{p(x)-2}\xi$ for all $\xi\in\mathbb{I}^N$;
 $p_{0,j}=\inf_{x\in\Omega_j}p(x)$ and
 $p_{1,j}=\sup_{x\in\Omega_j}p(x)$ for $j=1, 2$;
 $\Omega_1=\{ x\in\Omega; 1<p(x)<2\}$; and 
 $\Omega_2=\{ x\in\Omega;\quad 2\geq p(x)\}$.

Then, from \eqref{A: gp2}, \eqref{ml} and \eqref{F}, we conclude 
that $u_n$ converges strongly to $u*$ in $W_0^{1,p(x)}(\Omega)$. 
Similar argues allow to prove that  the sequence $v_n$ converges to $v^*$ 
strongly in $W_0^{1,q(x)}(\Omega)$.
\smallskip

\noindent\textbf{Step 3:} $(u^*,v^*)$ is a solution of \eqref{2} involving 
a fibering decomposition.
We show that $u^*={\bar r}{\bar z}$ and $v^*={\bar \rho}{\bar w}$ involve 
a solution of \eqref{2} via the fibering method. Let us recall that
  $\bar{z}$ and $\bar{w}$ are respectively the weak limit of $z_{n}$ and $w_n$
   is the weak limit of $w_{n}$.
The sequence $t_n=t(z_n,w_n)$ is defined as in \eqref{zp}. 
To simplify notation, we set  $r_n=t_n^{1/p^+}$ and 
$\rho_n=t_n^{1/q^+}$. Moreover, using \eqref{15} and \eqref{16},
 by extracting subsequences, if necessary, we can assume that 
$t_{n}$ converges in $\mathbb{I}$. We designate as  $\bar{t}=\lim_{n\to+\infty}t_n$. 
So, it follows $r_n\to {\bar t}^{1/p^+}$ and $\rho_n\to{\bar t}^{1/q^+}$ when 
$n$ tends to $+\infty$. We set  $\bar{r}=\bar{t}^{1/p^+}$ and 
$\bar{\rho}=\bar{t}^{1/q^+}$.

Because of the formulation 
 $$
\frac{1}{r_{n}}u_{n}-\frac{1}{\bar{r}} u^*
=\frac1{r_n{\bar r}}[({\bar r} -r_n)u^*+{\bar r} ( u_n-u^*)], 
$$
and the convergence results announced above,
  it is clear that 
$$
\| \frac{u_{n}}{r_{n}}-\frac{ u^*}{\bar{r}}\|_{1,p(x)} \to 0,\quad 
\text{as $n$ tends to } +\infty.
$$
In other words, since $\frac{u_{n}}{r_{n}}=z_n$, we deduce that $z_n$ 
converges strongly to $\frac{u^*}{\bar r}$ in $W_0^{1,p(x)}(\Omega)$.

 Thus, since $z_n$ converges weakly to ${\bar z}$ in $W_0^{1,p(x)}(\Omega)$, 
we deduce from above and also from uniqueness  $ \bar{z}=\frac{{u^*}}{\bar{r}}.$
On the other hand
$$
\| {u^*}\|_{1,p(x)}\leq \liminf_{n}\| u_{n}\|_{1,p(x)}
 \leq \limsup_{n}\| u_{n}\|_{1,p(x)}.
$$
So 
$$\|\bar{z} \|_{1,p(x)} \bar{r} \leq \liminf_{n}\| u_{n}\|_{1,p(x)} 
\leq \limsup_{n} \| u_{n}\|_{1,p(x)}
$$
thus
$$
\|\bar{z} \|_{1,p(x)}{r}\leq \liminf_{n}r_{n}\|{z}_{n}\|
\leq \limsup_{n} \| u_{n}\|_{1,p(x)}.
$$
Since $\| z_{n}\|_{1,p(x)}=1$ and 
$\| u_{n} \|_{1,p(x)} \leq  \| u_{n}- {u^*} \|_{1,p(x)}+\| {u^*}\|_{1,p(x)} ,$\\
we obtain
$$
\|\bar {z}\|_{1,p(x)}\bar{r} \leq  \bar{r}\leq \|\bar{z}\|_{1,p(x)}\bar{r}
$$
thus after dividing by ${\bar r>0}$, it occurs $\|\bar{z}\|_{1,p(x)}=1$.
In the same manner, we obtain
$\|\bar{w}\|_{1,q(x)}=1$.
We can conclude that $(\bar{z},\bar{w})$ is solution of the conditional  
problem \eqref{inftildeJ}. Furthermore, since 
$\|\bar{z}\|_{1,p(x)}=\|\bar{w}\|_{1,q(x)}=1$, using  \cite{xd}, we 
deduce the second part of  the Theorem \ref{E}.
The proof is complete.
\end{proof}

The material needed to prove  Theorem \ref{E} is complete. 
Next, we establish that the boundary value problem \eqref{2} 
admits at least one solution.

\subsection{Proof of Theorem \ref{E}}
  Existence of a critical point for ${\bf{J}}$.

\begin{proof}
The previous lemmas imply that $ (\bar{z},\bar {w} )$ is a conditional 
critical point for $\mathcal{J}$.
From the Euler-Lagrange characterization,  we deduce that there is a pair  
$(m_1,m_2)$ in $\mathbb{I}^2$ such that  for any $(h,k)\in X_0(x)$,
\begin{equation}\label{mm}
\nabla\mathcal{J}(\bar{z},\bar{w})\cdot (h,k)
= m_1\nabla{A}(\bar{z},\bar{w})\cdot (h,k)+m_2\nabla{B}(\bar{z},\bar{w})\cdot(h,k).
\end{equation}
In \eqref{mm}, we choose $h=\bar {z}$, $k=\bar{w}$, we obtain
\begin{equation}\label{Jzw}
 \mathcal{J}'(\bar{z},\bar{w})(\bar{z},\bar{w})=0.
 \end{equation}
Combining \eqref{mm} and \eqref{Jzw},  we obtain
\begin{gather*}
m_1A^{(1)}\cdot (\bar{z},\bar{w})+m_2 B^{(1)}\cdot (\bar{z},\bar{w})=0\\
m_1 {A}^{(2)}\cdot (\bar{z},\bar{w})+m_2 {B}^{(2)}\cdot (\bar{z},\bar{w})=0.
\end{gather*}
Here,  $A^{(1)}$, $B^{(1)}$ (resp. $A^{(2)}$ and $B^{(2)}$) denote the 
first  derivatives with respect to $z$ (resp. $w$).
Note taht 
\[\det \begin{pmatrix}
A^{(1)}\cdot (\bar{z},\bar{w}) &B^{(1)}\cdot(\bar{z},\bar{w})\\
A^{(2)}\cdot (\bar{z},\bar{w}) &B^{(2)}\cdot(\bar{z},\bar{w})
\end{pmatrix}
\geq p^{-}q^{-}A(\bar{z})B(\bar{w})=p^{-}q^{-}>0.
\]
It follows that $ m_1=m_2=0$.
Consequently, $ \mathcal{J}'(\bar{z},\bar{w})=0$,
or again,
$$ 
\mathbf{J}'(\bar{r}\bar{z},\bar{\rho}\bar{w}) =0
$$
Finally, we can conclude that $(u^*,v^*)=(\bar{r}\bar{z},\bar{\rho}\bar{w})$ 
is a critical point of $\mathbf{J}$.
\end{proof}
                       %%%%%%%%%%%%%%

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