\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
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)\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 $10$, $\theta <1$ 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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