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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 131, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/131\hfil Existence of solutions]
{Existence of solutions for  elliptic systems in
$\mathbb{R}^N$ involving the $p(x)$-Laplacian}

\author[A. Djellit,  Z. Youbi, S. Tas \hfil EJDE-2012/131\hfilneg]
{Ali Djellit, Zahra Youbi, Saadia Tas}  % in alphabetical order

\address{Ali Djellit \newline
 Mathematics, Dynamics and Modelization Laboratory\\
 Badji-Mokhtar Annaba University\\
 Annaba 23000, Algeria}
\email{a\_djellit@hotmail.com}

\address{Zahra Youbi \newline
 Mathematics, Dynamics and Modelization Laboratory \\
 Badji-Mokhtar Annaba University \\
 Annaba 23000, Algeria}
\email{zahra.youbi@yahoo.fr}

\address{Saadia Tas \newline
Applied Mathematics Laboratory \\
Abderrahmane Mira Bejaia University, Bejaia, Algeria}
\email{tas\_saadia@yahoo.fr}


\thanks{Submitted May 21, 2012. Published August 15, 2012.}
\subjclass[2000]{35J50, 35J92}
\keywords{$p(x)$-Laplacian operator; critical point, variational system}

\begin{abstract}
 This article  presents sufficient conditions for the existence of
 non-trivial solutions for a nonlinear elliptic system.
 To establish this result, we use a classical existence
 theorem in reflexive Banach spaces, under some growth
 conditions on the non-linearities.
\end{abstract}

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


\section{Introduction}

In this article  we establish the existence of nontrivial weak solution for
nonlinear elliptic system
\begin{equation}
\begin{gathered}
-\Delta _{p(x)}u=\frac{\partial F}{\partial u}(x,u,v)\quad\text{in }\mathbb{R}^N \\
-\Delta _{q(x)}v=\frac{\partial F}{\partial v}(x,u,v)\quad \text{in }\mathbb{R}^N
\end{gathered}  \label{e1}
\end{equation}
Here $p(x)$ and $q(x)$ are continuous real-valued functions such that
$1<p(x),q(x)<N$ $(N\geq 2)$ for all $x\in \mathbb{R}^N$.
The real-valued function $F$ belongs to
$C^{1}(\mathbb{R}^N\times \mathbb{R}^2)$, and $\Delta _{p(x)}$ is
the so-called $p(x)$-Laplacian operator; i.e., $\Delta _{p(x)}u=\operatorname{div}
(| \nabla u| ^{p(x)}\nabla u)$.

This decade bears witness to a considerable sum of results on non standard
growth conditions problems. This abundance is due to the recent
research developments in elasticity problems, electrorheological fluids,
image processing, flow in porous media, etc.; see for example
\cite{d1,o1}.

In a natural way, the introduction of the generalized
Lebesgue-Sobolev spaces turned out to be crucial \cite{d2,e1,f2}.
In this way, many authors could successfully deal with $p(x)$-Laplacian problems 
\cite{f1,f2}. 
Many additional works concern elliptic systems in relationship to standard and
nonstandard growth conditions. We refer the readers to  \cite{a1,f4,t2}
and the references therein.
In \cite{d3,t1}, the authors show the existence of nontrivial
solutions for the $(p,q)$-Laplacian system
\begin{gather*}
-\Delta _{p}u=\frac{\partial F}{\partial u}(x,u,v)\quad \text{in }\mathbb{R}^N \\
-\Delta _{q}v=\frac{\partial F}{\partial v}(x,u,v)\quad \text{in }\mathbb{R}^N
\end{gather*}
where the potential function $F$ satisfies mixed and subcritical growth 
conditions and, in addition, to be intimately connected with the first
eigenvalue of $p$-Laplacian operator. They apply the Mountain Pass 
theorem to obtain the nontrivial solutions of the system.

In \cite{e2}, the authors obtained the existence and multiplicity of
solutions for the vector valued elliptic system
\begin{gather*}
-\Delta _{p(x)}u=\frac{\partial F}{\partial u}(x,u,v)\quad \text{in }\Omega \\
-\Delta _{p(x)}v=\frac{\partial F}{\partial v}(x,u,v)\quad \text{in }\Omega \\
u=v=0\quad \text{on }\partial \Omega,
\end{gather*}
where $\Omega \subset \mathbb{R}^{N\text{ }}$ is a bounded domain with a
smooth boundary $\partial \Omega $, $N\geq 2$,
$(p,q)\in [C(\overline{\Omega })]^2$,
 $F\in C(\mathbb{R}^N\times \mathbb{R}^2,\mathbb{R})$.
Existence and multiplicity results are subjected to some natural growth 
conditions which guarantee the Mountain Pass geometry and
 Palais-Smale condition.

In \cite{x1}, the authors studied the  system
\begin{gather*}
-\Delta _{p(x)}u+| u| ^{p(x)-2}u=\frac{\partial F}{\partial u}(x,u,v)\quad
\text{in }\mathbb{R}^N \\
-\Delta _{p(x)}v+| v| ^{q(x)-2}v=\frac{\partial F}{\partial v}(x,u,v)\quad
\text{in }\mathbb{R}^N
\end{gather*}
The potential function $F$ needs to satisfy Caratheodory conditions.
Using critical point theory, they establish existence results in sub-linear and
super-linear cases.

In \cite{o1}, by the Mountain Pass theorem, the authors show the
existence of nontrivial solutions for the following $(p(x),q(x))$-Laplacian system
\begin{gather*}
-\Delta _{p(x)}u=\frac{\partial F}{\partial u}(x,u,v)\quad\text{in } \mathbb{R}^N \\
-\Delta _{q(x)}v=\frac{\partial F}{\partial v}(x,u,v)\quad \text{in }\mathbb{R}^N
\end{gather*}
where $F\in C^{1}(\mathbb{R}^N\times \mathbb{R}^2,\mathbb{R})$
verifies some mixed growth conditions.

With regard to  existence results, we use critical point theory. Our main
goal is to establish that the energy functional of the system is lower semi-
continuous and coercive in reflexive Banach space.

\section{Notation and hypotheses}

To discuss system \eqref{e1}, we recall some results on
generalized Lebesgue-Sobolev spaces.

Let $E(\Omega )$ be a space of functions defined on $\Omega $.
We set
$$
E_{+}(\Omega )=\{ h\in E(\Omega ):\inf_{x\in \Omega } h(x)>1\}.
$$
So, for all $h\in C_{+}(\mathbb{R}^N)$, we set
\begin{equation*}
h^{-}:=\inf_{x\in \mathbb{R}^N} h(x),\quad
h^{+}:=\sup_{x\in \mathbb{R}^N} h(x).
\end{equation*}
Let $M(\mathbb{R}^N)$ be the set of all measurable real-valued functions
defined on $\mathbb{R}^N$. For $p\in C_{+}(\mathbb{R}^N)$, we
designates the variable exponent Lebesgue space by
\begin{equation*}
L^{p(x)}(\mathbb{R}^N)=\{ u\in M(\mathbb{R}
^N):\int_{\mathbb{R}^N}| u(x)|
^{p(x)}dx<\infty \} ,
\end{equation*}
equipped with the so called Luxemburg norm
\begin{equation*}
| u| _{p(x)}:=| u| _{L^{p(x)}(\mathbb{R}^N)}
=\inf \{ \lambda >0:\int_{\mathbb{R}^N}| \frac{u(x)}{\lambda }| ^{p(x)}dx\leq
1\} .
\end{equation*}
This is a Banach space.
Define the Lebesgue-Sobolev space $W^{1,p(x)}(\mathbb{R}^N)$ by
\[
W^{1,p(x)}(\mathbb{R}^N)=\{ u\in L^{p(x)}(\mathbb{R}^N):| \nabla
u| \in L^{p(x)}(\mathbb{R}^N)\}, 
\]
equipped with the norm
\[
\| u\| _{1,p(x)}=\| u\|_{W^{1,p(x)}(\mathbb{R}^N)}
=|u| _{p(x)}+| \nabla u| _{p(x)}.
\]
The space $W_0^{1,p(x)}(\Omega )$ is
defined as the closure of $C_0^{\infty }(\Omega )$
in $W^{1,p(x)}(\Omega )$ with respect to
the norm $\| u\| _{1,p(x)}$. For $u\in W_0^{1,p(x)}(\Omega )$,
we can define an equivalent norm $\| u\| =| \nabla u|_{p(x)}$ ; 
since the well known Poincar\'e inequality holds.

Next, we recall some previous results. This way, we want to make the
proofs of the main results as transparent as possible.

\begin{proposition}[\cite{e1,f3}] \label{prop1}
If $p\in C_{+}(\mathbb{R}^N)$, then the spaces $L^{p(x)}(\mathbb{R}^N)$, 
$W^{1,p(x)}(\mathbb{R}^N)$ and $W_0^{1,p(x)}(\mathbb{R}^N)$
are separable and reflexive Banach spaces.
\end{proposition}


\begin{proposition}[\cite{e1,f3}] \label{prop2} 
The topological dual space of 
$L^{p(x)}(\mathbb{R} ^N)$ is $L^{p'(x)}(\mathbb{R}^N)$, where
\[
\frac{1}{p(x)}+\frac{1}{p'(x)}=1.
\]
Moreover for any $(u,v)\in L^{p(x)}(\mathbb{R}^N)\times L^{p'(x)}(\mathbb{R}^N)$,
we have
\[
| \int_{\mathbb{R}^N}uvdx| 
\leq (\frac{1}{p^{-}}+ \frac{1}{(p')^{-}})| u|_{p(x)}| v| _{p'(x)}
\leq 2| u| _{p(x)}| v|_{p'(x)}.
\]
\end{proposition}

Set $\rho (u)=\int_{\mathbb{R}^N}| u| ^{p(x)}dx$.

\begin{proposition}[\cite{e1,f3}] \label{prop3}
For all $u\in L^{p(x)}(\mathbb{R}^N)$, we have
\[
\min \{ | u| _{p(x)}^{p^{-}},| u| _{p(x)}^{p^{+}}\} 
\leq \rho (u)\leq \max \{ | u| _{p(x)}^{p^{-}},| u| _{p(x)
}^{p^{+}}\}.
\]
In addition, we have
\begin{itemize}
\item[(i)]  $| u| _{p(x)}<1$ (resp. $=1, >1$) $\Leftrightarrow \rho (u)<1$
(resp. $=1,>1$);

\item[(ii)] $| u| _{p(x)}>1 \Rightarrow | u| _{p(x)}^{p^{-}}\leq \rho (u)
 \leq | u| _{p(x)}^{p^{+}}$;

\item[(iii)] $| u| _{p(x) }>1\Rightarrow | u| _{p(x)}^{p^{+}}\leq \rho (u)
\leq | u| _{p(x)}^{p^{-}}$;

\item[(iv)] $\rho (\frac{u}{| u|_{p(x)}})=1$.
\end{itemize}
\end{proposition}


\begin{proposition}[\cite{e1}] \label{prop4} 
Let $p(x)$ and $q(x)$ be measurable functions such
that $p(x)\in L^{\infty }(\mathbb{R}^N)$ and
$1\leq p(x)q(x)\leq \infty $ almost every where in
$\mathbb{R}^N$. If $u\in L^{q(x)}(\mathbb{R}^N)$, $u\neq 0$.
Then
\begin{gather*}
| u| _{p(x)q(x)}\leq 1\Rightarrow |u| _{p(x)q(x)}^{p^{-}}
\leq \big|| u| ^{p(x)}\big| _{q(x)}\leq | u| _{p(x)q(x)}^{p^{+}},
\\
| u| _{p(x)q(x)}\geq 1\Rightarrow |u| _{p(x)q(x)}^{p^{+}}
\leq \big| | u| ^{p(x)}\big| _{q(x)}\leq | u| _{p(x)q(x)}^{p^{-}}.
\end{gather*}
In particular, if $p(x)=p$ is a constant, then
$| | u| ^{p}| _{q(x)}=|u| _{pq(x)}^{p}$.
\end{proposition}

\begin{proposition}[\cite{f3}] \label{prop5} 
If $u,u_{n}\in L^{p(x)}(\mathbb{R}^N)$,
 $n=1,2,\dots$, then the following statements are mutually equivalent:
\begin{itemize}

\item[(1)]  $\lim_{n\to \infty } | u_{n}-u|_{p(x)}=0$,

\item[(2)] $\lim_{n\to \infty } \rho (u_{n}-u)=0$,

\item[(3)] $u_{n}\to u$ in measure in $\mathbb{R}^N$ and 
$\lim_{n\to \infty } \rho (u_{n})=\rho (u)$.

\end{itemize}
\end{proposition}

Let $p^{\ast }(x)$ be the critical Sobolev exponent of $p(x)$ defined by
\[
p^{\ast }(x)=\begin{cases}
\frac{Np(x)}{N-p(x)} &\text{for } p(x)<N \\
+\infty &\text{for }p(x)\geq N
\end{cases}
\]
and let $C^{0,1}(\mathbb{R}^N)$ be the Lipschitz-continuous
functions space.

\begin{proposition}[\cite{d2,e1}] \label{prop6} 
If $p(x)\in C_{+}^{0,1}(\mathbb{R}^N)$, then
there exists a positive constant $c$ such that
\[
| u| _{p^{\ast }(x)}\leq c\|u\| _{p(x)},\quad \forall  
u\in W_0^{1,p(x)}(\mathbb{R}^N).
\]
\end{proposition}

Let $p\in L_{+}^{\infty }(\mathbb{R}^N)$ be an uniformly
continuous function such that $p^{+}<N$
and let $\Omega \subset $ $\mathbb{R}^N$ be a bounded domain.

\begin{proposition}[\cite{d2,e1}] \label{prop7}
(1) If $q\in L_{+}^{\infty }(\mathbb{R}^N)$ and 
$p(x)\leq q(x)\ll p^{\ast }(x)$, for all $x\in \mathbb{R}^N$, then the embedding
$ W^{1,p(x)}(\mathbb{R}^N)\hookrightarrow L^{q(x)}(\mathbb{R}^N)$
is continuous but not compact.

(2) If $p$ is continuous on $\overline{\Omega }$ and $q$ is a measurable
function on $\Omega $, with $p(x)\ll q(x)\ll p^{\ast }(x)$, for all 
$x\in\Omega $, then the embedding
$W^{1,p(x)}(\Omega )\hookrightarrow L^{q(x)}(\Omega )$
is compact.
\end{proposition}

Observe that the solution of \eqref{e1} will belong to the product
space
\[
W_{p(x),q(x)}:=W_0^{1,p(x)}(\mathbb{R}^N)\times W_0^{1,q(x)}(\mathbb{R}^N),
\]
equipped with the norm
\[
\| (u,v)\| _{p(x)}=|\nabla u| _{p(x)}+| \nabla v|_{p(x)}.
\]
The space $W_{p(x),q(x)}'$ is the topological dual of $W_{p(x),q(x)}$ equipped 
with the usual dual norm.
For $(u,v)$ in $W_{p(x),q(x)}$, let us define the functionals $I$, $J$, $K$
\begin{gather*}
F(u,v)=\int_{\mathbb{R}^N}F(x,u(x),v(x))dx,\\
J(u,v)=\int_{\mathbb{R}^N}\frac{1}{p(x)}| \nabla
u| ^{p(x)}dx+\int_{\mathbb{R}^N}\frac{1}{q(x)}| \nabla v| ^{q(x)}dx, \\
I(u,v)=J(u,v)-F (u,v).
\end{gather*}


\subsection*{Hypotheses}
We assume some growth conditions:
\begin{itemize} 
\item[(H1)]  $F\in C^{1}(\mathbb{R}^N\times \mathbb{R}^2,\mathbb{R})$ 
and $F(x,0,0)=0$.

\item[(H2)] There exist positive functions $a_i,b_i $ such that
\begin{gather*}
| \frac{\partial F}{\partial u}(x,u,v)| \leq a_1(x)| u|
^{p_1^{-}-1}+a_2(x)| v| ^{p_1^{+}-1}, \\
| \frac{\partial F}{\partial v}(x,u,v)|
\leq b_1(x)| u|^{q_1^{-}-1}+b_2(x)| v| ^{q_1^{+}-1},
\end{gather*}
where
$1<p_1(x),q_1(x)<\inf (p(x),q(x))$,  and
$p(x)$, $q(x)>\frac{N}{2}$, for all $x\in \mathbb{R}^N$.
The weight-functions $a_i$ and $b_i$, $i=1,2$, belong respectively to
the generalized Lebesgue spaces $L^{\alpha _i}(\mathbb{R}^N)$ and 
$L^{\beta }(\mathbb{R}^N)$, where
\[
\alpha _1(x)=\frac{p(x)}{p(x)-1},\beta (x)
=\frac{p^{\ast }(x)q^{\ast }(x)}{p^{\ast }(x)q^{\ast }(x)-p^{\ast
}(x)-q^{\ast }(x)},\quad
\alpha _2(x)=\frac{q(x)}{q(x)-1}.
\]

\item[(H3)] There exist constants $R>0,\theta >0$, and a
positive function
$H:\mathbb{R}^N\times \mathbb{R}^2\to \mathbb{R}$ such that for 
$x\in \mathbb{R}^N$, $| u| ,| v| \leq R$ and $t>0$ sufficiently
small, we have
\[
F(x,t^{1/p(x)}u,t^{1/q(x)}v)\geq t^{\theta }H(x,u,v).
\]
\end{itemize}

Assumption (H3) implies that the potential function $F$ is sufficiently
positive in a neighborhood of zero.



\begin{lemma} \label{lem1}
Under assumptions {\rm (H1)--(H2)},  the functional $F$
is well  defined and Frechet differentiable. Its derivative is
\[
F '(u,v)(\omega ,z)=\int_{\mathbb{R}^N}\frac{\partial F}{\partial u}(x,u,v)\omega
 +\frac{\partial F }{\partial v}(x,u,v)zdx,\forall (u,v),(\omega ,z)\in W_{p(x),q(x)}.
\]
\end{lemma}

\begin{proof} The functional $F $ is well defined on $W_{p(x),q(x)}$. 
Indeed, for all pair of real-valued functions $(u,v)\in W_{p(x),q(x)}$, 
we have in virtue of (H1) and (H2),
\begin{align*}
F(x,u,v)
&=\int_0^u\frac{\partial F}{\partial s}(x,s,v)ds+F(x,0,v)\\
&=\int_0^u \frac{\partial F}{\partial s}(x,s,v)ds+\int_0^{v}\frac{
\partial F}{\partial s}(x,0,s)ds+F(x,0,0).
\end{align*}
Then
\begin{equation}
F(x,u,v)\leq c_1[a_1(x)|u| ^{p_1^{-}}
+a_2(x)| v|^{p_1^{+}-1}| u| +b_2(x)|v| ^{q_1^{+}}]  \label{e2}
\end{equation}
Since $W^{1,p(x)}(\mathbb{R}^N)\hookrightarrow L^{s(x)p(x)}(\mathbb{R}^N)$ for 
$s(x)>1$, we have
\[
\big| | u| ^{p_1^{-}}\big|_{p(x)}=| u| _{p_1^{-}p(x)}^{p_1^{-}}
\leq c\| u\| _{p(x)}^{p_1^{-}}.
\]
So, taking into account H\"older inequality, Propositions 
\ref{prop2}, \ref{prop4}, \ref{prop6}, \ref{prop7} and 
(H2), we obtain
\begin{align*}
F (u,v)
&=\int_{\mathbb{R}^N} F(x,u,v)\,dx\\
&\leq c_2\Big(| a_1| _{\alpha _1(x)}| u| _{p_1^{-}p(x)}^{p_1^{-}}
 +| a_2| _{\beta (x)}| v| _{(p_1^{+}-1)
 q^{\ast }(x)}^{p_1^{+}-1}| u| _{p^{\ast }(x)}
+| b_2| _{\alpha _2(x) }| v| _{q_1^{+}q(x)}^{q_1^{+}}\Big)\\
&\leq c_{3}(| a_1| _{\alpha _1(x)}\|u\| _{p(x)}^{p_1^{-}}
 +| a_2| _{\beta (x)}\| v\| _{q(x)}^{p_1^{+}-1}\| u\| _{p(x)}
 +|b_2| _{\alpha _2(x)}\| v\|_{q(x)}^{q_1^{+}})
<\infty 
\end{align*}
The proof  is complete.
\end{proof}

Similarly, we show that $F'$ is also well defined.
Indeed, for all $(u,v),(\omega ,z)\in W_{p(x),q(x)}$, we can write
\begin{align*}
F '(u,v)(\omega ,z)
&=\int_{\mathbb{R}^N}\frac{\partial F}{\partial u}(x,u,v)\omega
dx+\int_{\mathbb{R}^N}\frac{\partial F}{\partial v}(x,u,v)
z\,dx \\
&\leq \int_{\mathbb{R}^N}( a_1(x)| u| ^{p_1^{-}-1}
+a_2( x)| v| ^{p_1^{+}-1})\omega\, dx\\
&\quad + \int_{\mathbb{R}^N}(b_1(x)| u| ^{q_1^{-}-1}
+b_2(x)| v| ^{q_1^{+}-1})z\,dx
\end{align*}
Following H\"{o}lder inequality, we obtain
\begin{align*}
F '(u,v)(\omega ,z)
&\leq c_4(| a_1| _{\alpha _1(x)}| |u| ^{p_1^{-}-1}| _{p^{\ast }(x)}| 
\omega | _{p(x)}+|a_2| _{\beta (x)}| | v|^{p_1^{+}-1}| _{q^{\ast }(x)}| \omega
| _{p^{\ast }(x)}\\
&\quad + | b_1|_{\beta (x)}| | u| ^{q_1^{-}-1}| _{p^{\ast }(x)}|z| _{q^{\ast }(x)}
+| b_2|_{\alpha _2(x)}| | v|^{q_1^{+}-1}| _{q^{\ast }(x)}|z| _{q(x)})
\end{align*}
The above propositions yield
\begin{align*}
F '(u,v)(\omega ,z)
&\leq c_{5}(| a_1| _{\alpha _1(x)}\| u\| _{p(x)}^{p_1^{-}-1}\| \omega \|_{p(x)}
+| a_2| _{\beta (x) }\| v\| _{q(x)}^{p_1^{+}-1}\| \omega\| _{p(x)}\\
&\quad +  | b_1|_{\beta (x)}\| u\| _{p(x)}^{q_1^{-}-1}\| z\| _{q(x)}
+|b_2| _{\alpha _2(x)}\| v\|_{q(x)}^{q_1^{+}-1}\| z\| _{q(x)})<\infty .
\end{align*}
Moreover $F $ is Frechet differentiable; namely, for any fixed
point $(u,v)\in W_{p(x),q(x)}$, and for any
$\varepsilon >0$, there exist $\delta =\delta _{\varepsilon ,u,v}>0$ such
that for all $(\omega ,z)\in W_{p(x),q(x)}$,
satisfying $\| (\omega ,z)\| _{p(x),q(x)}<\delta $ we have
\[
| F (u+\omega ,v+z)-F (u,v)-F '(u,v)(\omega ,z)| 
\leq \varepsilon \| (\omega ,z)\|_{p(x),q(x)}.
\]
First, let $B_R$ be the ball in $\mathbb{R}^N$ centered at the origin
and of radius $R$. Set
$B_R'=\mathbb{R}^N-B_R$.

It is well-known that the functional $F _R$ defined on 
$W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R)$
by
\[
F _R(u,v)=\int_{B_R}F(x,u,v)dx
\]
belongs to $C^{1}(W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R))$, by
in virtue of (H1) and (H2).
In addition, the operator $F _R'$ defined from 
$W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R)$ to
$(W_0^{1,p(x)}(B_R)\times W_0^{1,p(x)}(B_R))'$ by
\[
F _R'(u,v)(\omega ,z)
=\int_{B_R}\frac{\partial F}{\partial u}(x,u,v)\omega +\frac{\partial F}{
\partial v}(x,u,v)zdx,
\]
is compact (see \cite{f3}).
Clearly, for all $(u,v),(\omega ,z)\in W_{p(x),q(x)}$, we can write
\begin{align*}
&| F (u+\omega ,v+z)-F (u,v)-F '(u,v)(\omega ,z)| \\
&\leq  | F _R(u+\omega ,v+z)-F _R(u,v)-F_R'(u,v)(\omega ,z)| \\
&\quad + | \int_{B_R'}(F(x,u+\omega ,v+z)-F(x,u,v))-
\frac{\partial F}{\partial u}(x,u,v)\omega -\frac{\partial F}{
\partial v}(x,u,v)zdx|
\end{align*}
According to a classical theorem, there exist
 $\zeta _1,\zeta _2\in ] 0,1[ $, such that
\begin{align*}
&\big| \int_{B_R'}(F(x,u+\omega ,v+z)-F(x,u,v))
-\frac{\partial F}{\partial u}(x,u,v)\omega 
-\frac{\partial F}{\partial v}(x,u,v)zdx\big|\\
& =\big| \int_{B_R'}\frac{\partial F}{\partial u}(
x,u+\zeta _1\omega ,v)\omega +\frac{\partial F}{\partial v}(
x,u,v+\zeta _2z)z\\
&\quad -\frac{\partial F}{\partial u}(x,u,v)
\omega -\frac{\partial F}{\partial v}(x,u,v)z\,dx\big|.
\end{align*}
Consequently, by  growth conditions (H2), we obtain
\begin{align*}
&| \int_{B_R'}(F(x,u+\omega ,v+z)
-F(x,u,v))-\frac{\partial F}{\partial u}(
x,u,v)\omega -\frac{\partial F}{\partial v}(x,u,v)z\,dx| \\
&\leq \int_{B_R'}a_1(x)(| u+\zeta _1\omega | ^{p_1^{-}-1}+| u|
^{p_1^{-}-1})\omega dx\\
&\quad +\int_{B_R'}a_2(x)(| v+\zeta _2z|^{p_1^{+}-1}+| v| ^{p_1^{+}-1})\omega \,dx\\
&\quad + \int_{B_R'}b_1(x)(| u+\zeta _1\omega | ^{q_1^{-}-1}+| u|
^{q_1^{-}-1})zdx\\
&\quad +\int_{B_R'}b_2(x) (| v+\zeta _2z| ^{q_1^{+}-1}+|v| ^{q_1^{+}-1})z\,dx.
\end{align*}

By an elementary inequality, Propositions \ref{prop4}, \ref{prop6} and the fact that
\begin{equation}
\begin{gathered}
| a_i| _{L^{p'(x)}(B_R')}\to 0,\quad | a_i|_{L^{\beta (x)}(B_R')}\to 0 \\
| b_i| _{L^{q'(x)}(B_R')}\to 0,\quad | b_i|_{L^{\beta (x)}(B_R')}\to 0,
\end{gathered}  \label{e3}
\end{equation}
for $R$ sufficiently large and $i=1,2$, we obtain the  estimate
\begin{align*}
&\big| \int_{B_R'}(F(x,u+\omega ,v+z)
-F(x,u,v)-\frac{\partial F}{\partial u}(x,u,v)
\omega -\frac{\partial F}{\partial v}(x,u,v)z)\,dx\big|\\
& \leq  \varepsilon (\| \omega \| _{p(x)}+\|z\| _{q(x)}).
\end{align*}
We prove now that $F '$ is continuous on $W_{p(x),q(x)}$. To this end, we let
$(u_{n},v_{n})\to (u,v)$ in $W_{p(x),q(x)}$ as $n\to \infty $. 
Then for any $(\omega ,z)\in W_{p(x),q(x)}$, we have
\begin{align*}
&| F '(u_{n},v_{n})(\omega,z)-F '(u,v)(\omega ,z)| \\
&\leq | F _R'(u_{n},v_{n})(\omega ,z)-F _R'(u,v)(\omega ,z)| 
+ | \int_{B_R'}\big(\frac{\partial F}{\partial u}(x,u_{n},v_{n})
-\frac{\partial F}{\partial u}(x,u,v)\big)\omega dx|\\
&\quad + | \int_{B_R'}\big(\frac{\partial F}{\partial v}(
x,u_{n},v_{n})-\frac{\partial F}{\partial v}(x,u,v)
\big)z\,dx|
\end{align*}
Note that
\[
| F _R'(u_{n},v_{n})(\omega ,z)-F _R'(u,v)(\omega ,z)| \to 0\quad
\text{as }n\to \infty ,
\]
since $F _R'$ is continuous on 
$W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R)$ (see \cite{f3}).
Using (H2) once again and \eqref{e2}, the other terms on the wrigth-hand side
of the above inequality tend to zero.


\begin{lemma} \label{lem2} 
Under assumptions {\rm (H1)--(H2)}, $F$ is lower weakly
semicontinuous in $W_{p(x),q(x)}$.
\end{lemma}


\begin{proof} 
Let $(u_{n},v_{n})$ be a weakly convergent sequence to $(u,v)$ in $W_{p(x),q(x)}$.
In the same way, we write
\[
| F (u_{n},v_{n})-F (u,v)| 
\leq | F _R(u_{n},v_{n})-F _R(u,v)|+| \int_{B_R'}(F(x,u_{n},v_{n})
-F(x,u,v))dx| 
\]
Since the restriction operator is continuous, the sequence 
$(u_{n},v_{n})$  is weakly convergent to $(u,v)$ in
 $W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R)$.
However $F _R$ is weakly lower semi-continuous. This result comes 
from growth conditions (H1) and (H2), and Sobolev compact inclusion
\[
W_0^{1,p(x)}(B_R)\times W_0^{1,q(x)}(B_R)\hookrightarrow L^{s(x)}(B_R)
\times L^{t(x)}(B_R),
\]
for all $(s,t)\in [p(x),p^{\ast }(x)[\times [q(x),q^{\ast }(x)[$.
Using \eqref{e2} and \eqref{e3}, both the terms on the
right-hand side of the last inequality tend to zero.
\end{proof}



We remark that the $C^{1}-$functional $J$ is weakly lower semi-continuous, 
and its derivative is given by
\[
J'(u,v)(\omega ,z)=\int_{\mathbb{R}^N}| \nabla u| ^{p(x)-2}\nabla u\nabla
\omega \,dx
+\int_{\mathbb{R}^N}| \nabla v| ^{q(x)-2}\nabla v\nabla z\,dx
\]
The Euler-Lagrange functional associated to the system \eqref{e1}
takes the form
\[
I(u,v)=\int_{\mathbb{R}^N}\frac{1}{p(x)}| \nabla u| ^{p(x)}
+\frac{1}{q(x)}| \nabla v| ^{q(x)}dx-\int_{\mathbb{R}^N}F(x,u,v)\,dx.
\]
In other words $\ I(u,v)=J(u,v)-F (u,v)$.
Observe that the weak solutions of the system \eqref{e1} are
precisely the critical points of the functional $I$.

\begin{lemma} \label{lem3} 
Under assumptions {\rm (H1)--(H2)}, the functional $I$
is coercive.
\end{lemma}

\begin{proof} 
We have
\begin{align*}
I(u,v)&=\int_{\mathbb{R}^N}\frac{1}{p(x)}| \nabla u| ^{p(x)}
+\frac{1}{q(x)}| \nabla v| ^{q(x)}-F(x,u,v)\,dx\\
&\geq \int_{\mathbb{R}^N}\frac{1}{p^{+}}|
\nabla u| ^{p(x)}+\frac{1}{q^{+}}| \nabla
v| ^{q(x)}\,dx\\
&\quad - \int_{\mathbb{R}^N}(a_1(x) | u| ^{p_1^{-}}+a_2(x)|
v| ^{p_1^{+}-1}| u| +b_2(x) | v| ^{q_1^{+}})dx\\
&\geq \frac{1}{p^{+}}\rho (\nabla u)+ \frac{1}{q^{+}}\rho (\nabla v)\\
&\quad -(| a_1| _{\alpha_1(x)}| | u|^{p_1^{-}}| _{p(x)}
+| a_2| _{\beta (x)}| | v| ^{p_1^{+}-1}| _{q^{\ast }(x)}| u| _{p^{\ast }(
x)}+| b_2| _{\alpha _2(x)}| | v| ^{q_1^{+}}| _{q(x)}).
\end{align*}
By Propositions \ref{prop3}, \ref{prop4}, \ref{prop6}
 and the Young inequality, we obtain
\begin{align*}
I(u,v) & \geq \frac{1}{p^{+}}\| u\| _{p(x)}^{p^{-}}
 +\frac{1}{q^{+}}\| v\| _{q(x)}^{q^{-}}
-\Big(| a_1| _{\alpha_1(x)}\| u\| _{p(x)}^{p_1^{-}}\\
&\quad +| a_2| _{\beta (x)}(\frac{p_1^{+}-1}{p_1^{+}}\| v\| _{q(x)
}^{p_1^{+}}+\frac{1}{p_1^{+}}\| u\| _{p(x)
}^{p_1^{+}})+| b_2| _{\alpha _2(x)}\| v\| _{q(x)}^{q_1^{+}}\Big) \\
&\geq \frac{1}{p^{+}}\| u\| _{p(x)}^{p^{-}}
 +\frac{1}{q^{+}}\| v\| _{q(x)}^{q^{-}}
- c_{6}\Big(| a_1| _{\alpha_1(x)}\| u\| _{p(x)}^{p_1^{-}}\\
&\quad +| a_2| _{\beta (x)}\| v\| _{q(x)}^{p_1^{+}-1}
 +|a_2| _{\beta (x)}\| u\| _{p(x)}+| b_2| _{\alpha _2(x)
}\| v\| _{q(x)}^{q_1^{+}}\Big)
\end{align*}
Clearly, $I(u,v)$ tends to infinity as $\| (u,v)\| _{p(x),q(x)}\to\infty $,
since $1<p_1(x),q_1(x)<\inf (p(x),q(x))$.
\end{proof}


\begin{theorem} \label{thm1} 
Under assumptions {\rm (H1)--(H3)}, the system \eqref{e1}
has a non-trivial weak solution.
\end{theorem}

\begin{proof} By lemmas \ref{lem1}, \ref{lem2} and \ref{lem3}, the functional $I$ is weakly lower
semi-continuous and coercive in $W_{p(x),q(x)}$.
Consequently, the functional $I$ has a global minimum
 (see \cite[Theorem 12]{s1}. On the other hand $I $ is $C^{1}$. Hence
this minimum is necessarily characterized by a critical point of $I$, which is
a weak solution of  \eqref{e1}. This solution is
nontrivial. Indeed, as $I(0,0)=0$, it is sufficient to show that there exists
 $(u_1,v_1)\in W_{p(x),q(x)}$ such that $I(u_1,v_1)<0$. 
Let $R>0$, $\theta <1$ and 
$(0,0)\neq (\varphi ,\psi )\in C_0^{\infty }(\mathbb{R}^N)\times C_0^{\infty }(
\mathbb{R}^N)$ with 
$| \varphi | ,|\psi | \leq R$. According to (H3), one has
\begin{align*}
&I(t^{1/p(x)}\varphi ,t^{1/q(x)}\psi )\\
&=J(t^{1/p(x)}\varphi ,t^{1/q(x)}\psi )-F (t^{1/p(x)}\varphi ,t^{1/q(x)}\psi )
\\
&\leq t\int_{\mathbb{R}^N}[\frac{1}{p^{-}}| \nabla \varphi | ^{p(x)}
+\frac{1}{q^{-}} | \nabla \psi | ^{q(x)}] dx
-\int_{\mathbb{R}^N}F(x,t^{1/p(x)}\varphi ,t^{1/q(x)}\psi )dx
\\
&\leq t[\frac{1}{p^{-}}\rho (\nabla \varphi )
 +\frac{1}{q^{-}}\rho (\nabla \psi )]
-t^{\theta }\int_{\mathbb{R}^N}H(x,\varphi ,\psi )dx
\\
&\leq t[\frac{1}{p^{-}}\max \{ |\nabla \varphi | _{p(x)}^{p^{-}},| \nabla
\varphi | _{p(x)}^{p^{+}}\} 
+\frac{1}{q^{-}} \max \{ | \nabla \psi | _{q(x)}^{q^{-}},| \nabla \psi | _{q(x)
}^{q^{+}}\} ] 
\\
&\quad - t^{\theta }\int_{\mathbb{R}^N}H(x,\varphi ,\psi )dx
\\
&\leq t[\frac{1}{p^{-}}\max \{ \| \nabla \varphi \| _{p(x)}^{p^{-}},\| \nabla
\varphi \| _{p(x)}^{p^{+}}\} +\frac{1}{q^{-}}
\max \{ \| \nabla \psi \| _{q(x)}^{q^{-}},\| \nabla \psi \| _{q(x)
}^{q^{+}}\} ]\\
&\quad - t^{\theta }\int_{\mathbb{R}^N}H(x,\varphi ,\psi )\,dx<0,
\end{align*}
for $t>0$ sufficiently small.
\end{proof}

\subsection*{Acknowledgements}
This work was partially supported by PNR (ANDRU) contracts.

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\end{document}