\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 144, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/144\hfil Multiple solutions]
{Multiple solutions for a quasilinear\\ $(p,q)$-elliptic system}

\author[S. M. Khalkhali, A. Razani \hfil EJDE-2013/144\hfilneg]
{Seyyed Mohsen Khalkhali, Abdolrahman Razani}  % in alphabetical order

\address{Seyyed Mohsen Khalkhali \newline
Department of Mathematics, Science and Research branch,
Islamic Azad University, Tehran, Iran}
\email{sm.khalkhali@srbiau.ac.ir}

\address{Abdolrahman Razani \newline
Department of Mathematics, Imam Khomeini International University,
Qazvin, Iran}
\email{razani@ikiu.ac.ir}

\thanks{Submitted May 8, 2013. Published June 25, 2013.}
\subjclass[2000]{35J50, 35D30, 35J62, 35J92, 49J35}
\keywords{Weak solutions; critical points; Dirichlet system;
\hfill\break\indent divergence type operator}

\begin{abstract}
 We prove the existence of three weak solutions of a quasilinear
 elliptic system involving a general $(p, q)$-elliptic operator 
 in divergence form, with $1 < p \leqslant n$, $1 < q \leqslant n$. 
 Our main tool is an adaptation of a three critical points
 theorem due to Ricceri.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth
boundary $\partial\Omega$ and $1<p\leqslant n$, $1<q\leqslant n$.
In this article, we show the existence of multiple solutions for
system of elliptic differential equations
\begin{equation} \label{p}
\begin{gathered}
-\operatorname{div}(a_1(x,\nabla u))
=\lambda  g_1(x,u)+\mu F_u(x,u,v)\quad\text{in } \Omega\\
-\operatorname{div}(a_2(x,\nabla v))=\lambda  g_2(x,v)+\mu F_v(x,u,v)
\quad\text{in } \Omega\\
u=0,\quad v=0\quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $1<p,q\leqslant n$.

Many publication, such as \cite{BF,Bo, CC}, discuss
quasilinear elliptic systems involving $p$-Laplacian
operators and show the existence and multiplicity of solutions.
Boccardo and Figueiredo \cite{BF}
studied the existence of solutions for
\begin{gather*}
-\Delta_pu=F_u(x,u,v)\quad\text{in } \Omega\\
-\Delta_qu=F_v(x,u,v)\quad\text{in } \Omega\\
 u=0,\quad v=0\quad \text{on }\partial\Omega
\end{gather*}
where $p,q$ are real numbers larger than 1.

Using the fibering method introduced by Pohozaev,  Bozhkov and
Mitidieri \cite{Bo} proved the existence
of multiple solutions for a quasilinear system involving a pair of
 ($p$,$q$)-Laplacian operators.
In \cite{CC} the existence of three solutions for the eigenvalue problem
\begin{equation}\label{eq.15}
\begin{gathered}
-\Delta_pu=\lambda F_u(x,u,v)\quad\text{in } \Omega\\
-\Delta_qu=\lambda F_v(x,u,v)\quad\text{in } \Omega\\
 u=0,\quad v=0\quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $p>n,q>n$ is ensured for suitable $F$.

Some other works \cite{Br,K,DV,NM} studied mainly  problems involving
 $p$-Laplacian type elliptic operators in divergence form and related
eigenvalue problems
\begin{gather*}
-\operatorname{div}(a(x,\nabla u))=\lambda f(x,u)\quad\text{in } \Omega\\
u=0\quad \text{on }\partial\Omega
\end{gather*}
These operators have $p$-Laplacian operator as a simple case; i.e.,
 if $a(x,s)=|s|^{p-2}s$ then for $p\geqslant 2$ we have
$\Delta_pu=\operatorname{div}(a(x,\nabla u))$ and moreover they have other important cases,
such as the generalized mean curvature operator
$\operatorname{div}\big((1+|\nabla u|^2)^\frac{p-2}{2}\nabla u\big)$
which is generated by $a(x,s)=(1+|s|^2)^\frac{p-2}{2}s$ and is
used in studying the geometric properties of manifolds especially minimal
surfaces.

The existence of multiple solutions for this type of nonlinear differential
equations was studied in \cite{DP,K}.
 Many of these results are based on some three critical points theorems
of Ricceri and Bonanno established
 in \cite{R1,B1}. In \cite{R3}, Ricceri developed one of his results,
\cite[Theorem 1]{R1} by means of an abstract result,
\cite[Theorem 4]{R2}.

In this article, we shall give a variant of Ricceri's three critical
points theorem \cite{R3} which it seems its verification for some
type of elliptic operators like $\operatorname{div}\big(a(x,\nabla u)\big)$
is easier. As an application, we study the
existence of at least three weak solutions for \eqref{p}.
Our approach in dealing with \eqref{p} is very close to Ricceri's
one in \cite{R3} but employs some calculations
of \cite{NM} to adjust it to our problem.

\section{Preliminaries}

In the sequel, for any $\xi=(\xi_1,\xi_2,\ldots,\xi_n)\in\mathbb{R}^n$
by $|\xi|$ we mean the usual Euclidean norm of $\xi$; that is,
$|\xi|=\sqrt{\xi_1^2+\xi_2^2+\cdots+\xi_n^2}$
which is produced by the inner product $\xi\cdot\eta=\sum_{i=1}^n\xi_i\eta_i$
in which $\xi,\eta\in\mathbb{R}^n$. Also
for every $1\leqslant p<\infty$ and open $\Omega\subset\mathbb{R}^n$ and
measurable $u:\Omega\rightarrow\mathbb{R}$ we define
\[
\|u\|_{L^p(\Omega)}=\Big(\int_{\Omega}|u|^pdx\Big)^{1/p}
\]
 and for $p>1$ we assume the reflexive separable Sobolev space
 $W_0^{1,p}(\Omega)$  is endowed with the norm
\[
\|u\|_p=\Big(\int_\Omega \vert \nabla u\vert^p dx\Big)^{1/p}
\]
 which is equivalent with its usual norm
\[
\|u\|_{W_0^{1,p}(\Omega)}
    =\Big(\int_\Omega \vert u\vert^p+\vert \nabla u\vert^p dx\Big)^{1/p}.
\]
By setting $p_1=p$, $p_2=q$, and inspired by De N\'{a}poli
and Mariani \cite{NM} and Deng and Pi \cite{DP}, we assume that the
$a_i:\overline{\Omega}\times\mathbb{R}^n\to\mathbb{R}^n$, for $i=1,2$, satisfy
the following conditions:
\begin{itemize}
\item[(H1)] There exists continuous function
$A_i:\overline{\Omega}\times\mathbb{R}^n\to\mathbb{R}$ such that
 $A_i(x,\xi)$ has $a_i(x,\xi)$ as its continuous derivative with respect
to $\xi$ at every $(x,\xi)\in\overline{\Omega}\times\mathbb{R}^n$
with the following additional properties:
\begin{itemize}

\item[(a)] $A_i(x,0)=0,\quad \forall x\in\Omega$.

\item[(b)] There exists some constant $C_1>0$ such that $a_i$ satisfies the growth condition
\begin{equation}
|a_i(x,\xi)|\leqslant C_1 (1+|\xi|^{p_i-1}),\quad\forall\xi\in\mathbb{R}^n.
\label{Hb}
\end{equation}

\item[(c)] $A_i$ is strictly convex: For every $t\in [0,1]$
\begin{equation}\label{eq.4}
A_i\big(x,(1-t)\xi+t\eta\big)\leqslant(1-t)A_i(x,\xi)+t A_i(x,\eta),\quad\forall x\in\Omega,\ \forall\xi,\eta\in\mathbb{R}^n
\end{equation}
and this inequality is strict if $t\in(0,1)$.

\item[(d)] $A_i$ satisfies the ellipticity condition: There exists a constant $C_2>0$ such that
\begin{equation}\label{He}
A_i(x,\xi)\geqslant C_2|\xi|^{p_i},\quad\forall x\in\Omega,\ \forall\xi\in\mathbb{R}^n.
\end{equation}
\end{itemize}
\end{itemize}
Assumption (H1) has some consequences that will be
helpful in this article.  From the strict convexity and
differentiability of $A_i(x,\xi)$ with respect to $\xi$, and
assumption (H1)(c),  we have
\[
A_i(x,\eta)\geqslant A_i(x,\xi)+a_i(x,\xi)(\eta-\xi),
\]
from which it follows that
\begin{equation}\label{eq.3}
\big(a_i(x,\xi)-a_i(x,\eta)\big)\cdot(\xi-\eta)\geqslant 0,
\end{equation}
for every $x\in\Omega$ and $\xi,\eta\in\mathbb{R}^n$.
Also, from \eqref{eq.3} we obtain
\begin{equation}\label{eq.5}
a_i(x,\xi+t\eta)\eta\geqslant a_i(x,\xi)\eta
\end{equation}
for every $t>0$ and $\xi,\,\eta\in\mathbb{R}^n$.

We say the mapping $F:X\to X^*$ satisfies the $S_+$ condition,
if every sequence $\{x_n\}_{n=1}^\infty$ in $X$ such that $x_n\rightharpoonup x$
and $\limsup_{n\to\infty}\langle F(x_n),x_n-xt\rangle\leqslant 0$
has a convergent subsequence $\{x_{n_k}\}_{k=1}^\infty$ such that
$x_{n_k}\to x$.

\begin{proposition}\label{prop1}
 Let $X$ be a reflexive Banach space and $F,J:X\to\mathbb{R}$
two $C^1$ functionals on $X$. If the mapping $F':X\to X^*$ satisfies $S_+$
condition and $J':X\to X^*$ is compact and $F+J:X\to\mathbb{R}$ is
coercive then $F+J$ satisfies the Palais-Smale condition.
\end{proposition}

\begin{proof}
 If $\{x_n\}_{n=1}^\infty$ is a sequence in $X$ such that
$|F(x_n)+J(x_n)|<M$ for some $M>0$ and any $n\in\mathbb{N}$ and
$\|F'(x_n)+J'(x_n)\|\to 0$ then coercivity of $F+J$ implies boundedness
of $\{x_n\}_{n=1}^\infty$ and since $X$ is reflexive, there exists a
subsequence $\{x_{n_k}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$ and
$x\in X$ such that $x_{n_k}\rightharpoonup x$. Now compactness of
$J':X\to X^*$ implies there exists $x^*\in X^*$ such that $J(x_{n_k})\to x^*$
up to a subsequence. Then since
\[
\langle J'(x_{n_k}),x_{n_k}-x\rangle
=\langle J'(x_{n_k})-x^*,x_{n_k}-x\rangle+\langle x^*,x_{n_k}-x\rangle
\]
and $\{x_{n_k}\}_{k=1}^\infty$ is bounded and $x_{n_k}\rightharpoonup x$,
we have $\langle J'(x_{n_k}),x_{n_k}-x\rangle\to 0$. Therefore,
\begin{align*}
&\limsup_{n\to\infty}\langle F'(x_{n_k}),x_{n_k}-x\rangle\\
&\leqslant\limsup_{n\to\infty}\langle F'(x_{n_k})+J'(x_{n_k}),x_{n_k}-x\rangle
 -\lim_{n\to\infty}\langle J'(x_{n_k}),x_{n_k}-x\rangle\\
&\leqslant\limsup_{n\to\infty}\|F'(x_{n_k})+J'(x_{n_k})\|\,\|x_{n_k}-x\|=0.
\end{align*}
Hence, by $S_+$ condition of $F'$, for a subsequence of
$\{x_{n_k}\}_{k=1}^\infty$ without relabeling $x_{n_k}\to x$.
\end{proof}


\section{Main results}

First we give a theorem that is a variant of \cite[Theorem 1]{R3}. 

\begin{theorem}\label{thm1}
 Let $X$ be a separable and reflexive real Banach space;
$I\subset\mathbb{R}$ an interval; $\Phi:X\to\mathbb{R}$ a weakly
sequentially lower semicontinuous $C^1$ functional, bounded on each bounded
subset of $X$ and has unique global minimum at $x_0\in X$ and further
the mapping $\Phi':X\to X^*$ satisfies $S_+$ condition and for every bounded
$E\subset X$ there exist constants $C>0$ and $\nu>0$ such that for every
$x\in E$
\[
\Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.
\]
Also suppose $J:X\to\mathbb{R}$ be a $C^1$ functional with compact derivative
such that for each $\lambda\in I$, the functional $\Phi-\lambda J$ is coercive
and has a strict local not global minimum at $x_0$.

Then for each compact interval $[a,b]\subset I$, there exists $r>0$
with the following property:
for every $\lambda\in[a,b]$ and every $C^1$ functional $\Psi:X\to\mathbb{R}$
with compact derivative, there exists $\delta>0$ such that,
for each $\mu\in[0,\delta]$, the equation
\[
\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)
\]
has at least three solutions whose norms are less than $r$.
\end{theorem}

To prove the above theorem, we need the following lemma which
is a variant of \cite[Theorem C]{R3}.

\begin{lemma}\label{lem5}
 Let $X$ be a separable and reflexive real Banach space,
 $\Phi:X\to\mathbb{R}$ a functional that has unique global minimum at
$x_0\in X$ and furthermore for every bounded $E\subset X$ there exist
constants $C>0$ and $\nu>0$ such that for every $x\in E$
\begin{equation}\label{eq.1}
\Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.
\end{equation}
 Let $J:X\to\mathbb{R}$ be a weakly sequentially lower semicontinuous
functional. Assume that $\Phi+J$ has a local strict minimum at $x_0$ in
the strong topology of $X$ and
\[
 \lim_{\|x\|\to\infty}\big(\Phi(x)+J(x)\big)=\infty.
\]
Then $x_0$ is a strict local minimum of $\Phi+J$ in the weak topology of $X$.
\end{lemma}

\begin{proof}
The main part of the proof is the same as that of \cite[Theorem C]{R3}.
We show $x_0$ must be a strict local minimum in the weak topology of $X$.
 If not, by assumption there exists $\rho>0$ such that
\[
\Phi(x_0)+J(x_0)<\Phi(x)+J(x)
\]
for every $x\in X$ satisfying $\|x\|>\rho$. Set
\[
B=\{x\in X\ : \|x\|\leqslant\rho\}.
\]
Since $X$ is separable and reflexive, the set $B$ is metrizable in its weak
topology which we denote its metric by $\sigma$.
Since we suppose $x_0$ is not a strict local minimum in weak topology of $X$,
there exists a sequence $\{x_n\}$ in $X$ such that for every $n\in\mathbb{N}$,
\begin{equation}\label{eq.7}
\sigma(x_0,x_n)<\frac{1}{n},\quad
\Phi(x_n)+J(x_n)\leqslant\Phi(x_0)+J(x_0).
\end{equation}
So, $x_n\in B$ and $x_n\rightharpoonup x_0$. Then weakly sequentially
lower semicontinuity of $J$ implies
\begin{align*}
\liminf_{n\to\infty}\Phi(x_n)+J(x_0)
&\leqslant\liminf_{n\to\infty}\Phi(x_n) +\liminf_{n\to\infty}J(x_n)\\
&\leqslant\liminf_{n\to\infty}\big(\Phi(x_n)+J(x_n)\big)
\leqslant\Phi(x_0)+J(x_0).
\end{align*}
and therefore,
\[
\liminf_{n\to\infty}\Phi(x_n)\leqslant\Phi(x_0).
\]
But $\Phi(x_0)$ is the global minimum of $\Phi(x)$ so, for a suitable convergent
subsequence of $\Phi(x_n)$ we have
\[
\lim_{n\to\infty}\Phi(x_n)=\Phi(x_0)
\]
then by \eqref{eq.1} we have $x_n\to x_0$ which contradicts strict local
minimality of $\Phi(x_0)+J(x_0)$ in the strong topology of $X$ by \eqref{eq.7}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
 Following the arguments in \cite[Theorem 1]{R3}, since any $C^1$ functional
with compact derivative on $X$ is weakly sequentially continuous
\cite[Corollary 41.9]{Z2}, and in particular, it is bounded on each bounded
subset of $X$, so for any compact $[a,b]\subset I$ and
$\sigma>\sup_{\lambda\in[a,b]}\big(\Phi(x_0)-\lambda J(x_0)\big)$,
\begin{align*}
&\cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)<\sigma\}\\
&\subset \{x\in X : \Phi(x)-a J(x)<\sigma\}
 \cup\{x\in X : \Phi(x)-b J(x)<\sigma\}.
\end{align*}
By the coercivity assumption, the set on the right is bounded and there exists
 $\eta>0$ such that
\begin{equation}\label{eq.8}
 \cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)<\sigma\}\subset B_{\eta}
\end{equation}
where $B_{\eta}=\{x\in X :\|x\|<\eta\}$. Now, set
\[
c^*=\sup_{B_{\eta}}\Phi+\max\{|a|,|b|\}\sup_{B_{\eta}}|J|
\]
and choose $r>\eta$ so that
\begin{equation}\label{eq.10}
 \cup_{\lambda\in[a,b]}\{x\in X : \Phi(x)-\lambda J(x)<c^*+2\}\subset B_r\,.
\end{equation}
Now, for any $C^1$ functional $\Psi:X\to\mathbb{R}$ with compact derivative,
choose a bounded $C^1$ function $g:\mathbb{R}\to\mathbb{R}$
with bounded derivative such that $g(t)=t$ for every
$-\sup_{B_r}|\Psi|\leqslant t\leqslant\sup_{B_r}|\Psi|$.
Then $\tilde{\Psi}:X\to\mathbb{R}$ defined by $\tilde{\Psi}(x)=g\circ\Psi(x)$
is a $C^1$ functional on $X$ such that $\tilde{\Psi}(x)=\Psi(x)$ for all
$x\in B_r$. On the other hand, for every $E\subset X$
\[
\tilde{\Psi}'(E)\subset g'\big(\Psi(E)\big)\Psi'(E)
\]
and therefore $\tilde{\Psi}':X\to X^*$ is compact. In addition,
by Lemma \ref{lem5} the functional $\Phi-\lambda J$ has a strict local
 minimum at $x_0$ in the weak topology of $X$, for any $\lambda\in[a,b]$.
 So, by applying \cite[Theorem 4]{R2} to the functionals $-\tilde{\Psi}$
and $\Phi-\lambda J$ by taking $\tau$ as the weak topology of $X$ and
considering \eqref{eq.8} and the fact that the topology $\tau_{\Phi-\lambda J}$
is weaker than the strong one, the existence of some $\gamma>0$ is deduced
such that for each $\mu\in[0,\gamma]$ the functional
$\Phi-\lambda J-\mu\tilde{\Psi}$
has at least two local minimum in $B_\eta$, say $x_1,x_2$. Now, If
\[
\delta=\min\{\gamma,\frac{1}{\sup_{\mathbb{R}}|g|}\}
\]
then for every $\mu\in[0,\delta]$ the functional
 $\Phi-\lambda J-\mu\tilde{\Psi}$ is coercive by assumption and satisfies
Palais-Smale condition, by Proposition \ref{prop1}. Set
\begin{gather*}
\mathcal{S}=\{u\in C([0,1],X): u(0)=x_1,\,u(1)=x_2\},\\
c_{\lambda,\mu}=\inf_{u\in\mathcal{S}}\sup_{t\in[0,1]}
\big(\Phi(u(t))-\lambda J(u(t))-\mu\tilde{\Psi}(u(t))\big)
\end{gather*}
then by the Mountain Pass Theorem \cite[Theorem 8.2]{AM}),
there exists $x_3\in X$ distinct from $x_1$ and $x_2$ such that
\[
\Phi'(x_3)-\lambda J'(x_3)-\mu\tilde{\Psi}'(x_3)=0,\quad
\Phi(x_3)-\lambda J(x_3)-\mu\tilde{\Psi}(x_3)=c_{\lambda,\mu}.
\]
Now since
\begin{align*}
 c_{\lambda,\mu}
&\leqslant\sup_{t\in[0,1]}\Phi(x_1+t(x_2-x_1))-\lambda J(x_1+t(x_2-x_1))
 -\mu\tilde{\Psi}(x_1+t(x_2-x_1))\\
&\leqslant\sup_{B_\eta}\Phi+\max\{|a|,|b|\}\sup_{B_\eta}J
 +\delta\sup_{\mathbb{R}}|g|\leqslant c^*+1,
\end{align*}
we have $\Phi(x_3)-\lambda J(x_3)< c^*+2$ and therefore
$x_3\in B_r$ by \eqref{eq.10}. Since $\Psi(x)=\tilde{\Psi}(x)$
for every $x\in B_r$ so $\Psi'(x_i)=\tilde{\Psi}'(x_i)$ for $i=1,2,3$.
Thus $x_1,x_2,x_3$ are three solutions of $\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)$
in $B_r$
\end{proof}

Our main tool in studying \eqref{p} is the following Theorem, which in fact
 is a restatement of \cite[Theorem 2]{R3}. It adopts it to our situation
and its proof is the same as that of \cite[Theorem 2]{R3},
except that we use Theorem \ref{thm1} instead of \cite[Theorem 1]{R3},
and remove the phrase $\hat{x}_\lambda=x_0$.
Therefore we omit its proof.

\begin{theorem}\label{thm2}
Let $X$ be a separable and reflexive real Banach space;
$I\subset\mathbb{R}$ an interval; $\Phi:X\to\mathbb{R}$ a weakly sequentially
lower semicontinuous $C^1$ functional that has unique global minimum
at $x_0\in X$ and for every bounded $E\subset X$ there exist some constants
$C>0$ and $\nu>0$ such that for every $x\in E$
\[
\Phi(x)-\Phi(x_0)\geqslant C\|x-x_0\|^\nu.
\]
Let $J:X\to\mathbb{R}$ be a $C^1$ functional with compact derivative.
Finally, setting
\[
\alpha=\max\big\{0,\,\limsup_{\|x\|\to\infty}\frac{J(x)}{\Phi(x)},
\,\limsup_{x\to x_0}\frac{J(x)}{\Phi(x)}\big\},\quad
\beta=\sup\big\{\frac{J(x)}{\Phi(x)}: x\in\Phi^{-1}(]0,\infty[)\big\},
\]
assume that $\alpha<\beta$.
Then, for each compact interval $[a,b]\subset ]\frac{1}{\beta},\frac{1}{\alpha}[$
 (with the conventions $\frac{1}{0}=\infty$, $\frac{1}{\infty}=0$) there exists
$r>0$ with the following property:
for every $\lambda\in[a,b]$ and every $C^1$ functional $\Psi:X\to\mathbb{R}$
with compact derivative, there exists $\delta>0$ such that, for each
$\mu\in[a,b]$, the equation
\[
\Phi'(x)=\lambda J'(x)+\mu\Psi'(x)
\]
has at least three solutions whose norms are less than $r$.
\end{theorem}

Hereafter we denote by $X$ the product real Banach space
$W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ in which $p,q>1$ and equip
it with the norm
\[
\|(u,v)\|=\|u\|_p+\|v\|_q=(\int_\Omega \vert \nabla u\vert^p dx)^{1/p}
+(\int_\Omega \vert \nabla v\vert^q dx)^\frac{1}{q}.
\]
At every $(u,v)\in X$, define
\begin{gather*}
\Phi(u,v)=\int_\Omega A_1(x,\nabla u)\,dx+\int_\Omega A_2(x,\nabla v)\,dx,\quad
\Psi(u,v)=\int_\Omega F\big(x,u(x),v(x)\big)\,dx,\\
J(u,v)=\int_\Omega\int_0^{u(x)}g_1(x,s)\,ds\,dx
 +\int_\Omega \int_0^{v(x)}g_2(x,s)\,ds\,dx
\end{gather*}
in which $g_1,g_2$ satisfy the following inequalities for some constant $C>0$,
\begin{equation}\label{eq.12}
|g_1(x,\xi)|\leqslant C(1+|\xi|^{\tau-1}),\quad
|g_2(x,\xi)|\leqslant C(1+|\xi|^{\kappa-1}),
\end{equation}
for a.e. $x\in\Omega$ where $1<\tau<p$ and $1<\kappa<q$.

Before stating and proving our main result for \eqref{p}, i.e.,
 Theorem \ref{thm3}, we establish some lemmas which are useful
in proving this theorem. In fact, we gathered needed hypotheses of
Theorem \ref{thm3} in these lemmas.

\begin{lemma}\label{lem1}
Let $\Phi:X\to\mathbb{R}$ be defined as above. If the functions
$A_i$ for $i=1,2$ satisfy  {\rm (H1)}, then $\Phi\in C^1(X;\mathbb{R})$.
In particular $\Phi':X\to X^*$ is continuous.
\end{lemma}

\begin{proof}
At $(u,v)\in X$ for every $(\xi,\mu)\in X$ and $0<|t|<1$, by applying the
Mean Value Theorem for $A_i$'s we obtain
\begin{align*}
&\langle\Phi'(u,v),(\xi,\mu)\rangle\\
&=\lim_{t\to 0}\frac{\Phi(u+t\xi,v+t\mu)-\Phi(u,v)}{t}\\
&=\lim_{t\to 0}\Big(\int_\Omega a_1(x, \nabla u+t\theta_1(x)
\nabla\xi)\cdot\nabla\xi\, dx+\int_\Omega a_2(x, \nabla v+t\theta_2(x)
\nabla\mu)\cdot\nabla\mu\, dx\Big)
\end{align*}
in which $0<\theta_1(x),\theta_2(x)<1$ for every $x\in\Omega$.
Now by the Cauchy-Schwarz  inequality and \eqref{Hb},
\begin{align*}
\big|a_1(x, \nabla u+t\theta_1(x)\nabla\xi)\cdot\nabla\xi\big|
&\leqslant C\big(1+|\nabla u+t\theta_1(x)\nabla\xi|^{p-1}\big)\,
|\nabla\xi|\\
&\leqslant C(1+2^{p-1}\big(|\nabla u|^{p-1}
+|\nabla\xi|^{p-1}\big))|\nabla\xi|,
\end{align*}
and since
\[
\int_\Omega(1+2^{p-1}\big(|\nabla u|^{p-1}+|\nabla\xi|^{p-1}\big))
|\nabla\xi|\,dx\leqslant C\Big(m(\Omega)+\|u\|_p^p+\|\xi\|_p^p\Big)
^{1/p'}\|\xi\|_p
\]
where $C$ denotes a constant and $m(\Omega)$ is the Lebesgue measure
of $\Omega$ and $p'=\frac{p}{p-1}$ is the H\"{o}lder conjugate of $p$,
then the Dominated Convergence Theorem implies
\[
\lim_{t\to 0}\int_\Omega a_1(x, \nabla u+t\theta_1(x)\nabla\xi)\cdot\nabla\xi\, dx
=\int_\Omega a_1(x, \nabla u)\cdot\nabla\xi\, dx.
\]
Similarly,
\[
\lim_{t\to 0}\int_\Omega a_2(x, \nabla v+t\theta_2(x)\nabla\mu)\cdot\nabla\mu\, dx
=\int_\Omega a_2(x, \nabla v)\cdot\nabla\mu\, dx,
\]
and the functional $\Phi$ is G\^{a}teaux differentiable at every
$(u,v)\in X$ and
\[
\langle\Phi'(u,v),(\xi,\mu)\rangle=\int_\Omega a_1(x,\nabla u)
 \cdot\nabla\xi+a_2(x,\nabla v)\cdot\nabla\mu\,dx.\quad \forall(\xi,\eta)\in X
\]
Now we prove $\Phi':X\to X^*$ is continuous. Suppose $(u_n,v_n)\to (u,v)$
in $X$ then by the H\"{o}lder inequality for every $(\xi,\eta)\in X$ we have
\begin{align*}
&\big|\langle\Phi'(u_n,v_n)-\Phi'(u,v),(\xi,\mu)\rangle\big|\\
&\leqslant\int_\Omega\Big|\Big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\Big)
 \cdot\nabla\xi\Big|+\Big|\Big(a_2(x,\nabla v_n)-a_2(x,\nabla v)\Big)
 \cdot\nabla\mu\Big|\,dx\\
&\leqslant\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}
 \|\xi\|_p+\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}\|\mu\|_q\\
&\leqslant\Big(\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}
 +\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}\Big)\|(\xi,\mu)\|,
\end{align*}
where $q'=\frac{q}{q-1}$ is the H\"{o}lder conjugate of $q$.
Hence, it is sufficient to show that
\[
\lim_{n\to\infty}\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}+\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}=0.
\]
If not, we have
\[
\limsup_{n\to\infty}\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}
+\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}>0,
\]
then there exists a subsequence of $\{(u_n,v_n)\}$ which we denote it
 by the same notation $\{(u_n,v_n)\}$ for which
\begin{equation}\label{eq.2}
\lim_{n\to\infty}\|a_1(x,\nabla u_n)-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}
+\|a_2(x,\nabla v_n)-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}>0.
\end{equation}
Since $(u_n,v_n)\to (u,v)$ in $X$, we have $u_n\to u$ and $v_n\to v$
in $W_0^{1,p}(\Omega)$ and $W_0^{1,q}(\Omega)$ respectively.
So there exist subsequences $\{u_{n_k}\}$ and $\{v_{n_k}\}$ of $\{u_n\}$
and $\{v_n\}$ respectively and some functions $g\in L^p(\Omega)$
and $h\in L^q(\Omega)$ such that $|\nabla u_{n_k}(x)|\leqslant g(x)$
and $\nabla u_{n_k}\to\nabla u\ \text{a.e.}$ and
$|\nabla v_{n_k}(x)|\leqslant h(x)$ and $\nabla v_{n_k}\to\nabla v$ a.e. as well.
Thus for some constant $C$ and a.e. $x\in\Omega$ we have
\[
|a_1(x,\nabla u_{n_k})-a_1(x,\nabla u)|
\leqslant C(2+|\nabla u_{n_k}|^{p-1}+|\nabla u|^{p-1})
\leqslant 2C(1+g^{p-1})
\]
and by a similar argument
\[
|a_2(x,\nabla v_{n_k})-a_1(x,\nabla v)|\leqslant 2C(1+h^{p-1}).
\]
Now by the Dominated Convergence Theorem
\[
\lim_{k\to\infty}\|a_1(x,\nabla u_{n_k})-a_1(x,\nabla u)\|_{L^{p'}(\Omega)}
+\|a_2(x,\nabla v_{n_k})-a_2(x,\nabla v)\|_{L^{q'}(\Omega)}=0,
\]
which contradicts \eqref{eq.2}. Therefore $\Phi':X\to X^*$ is continuous
and \emph{a priori} $\Phi\in C^1(X;\mathbb{R})$.
\end{proof}

\begin{lemma}\label{lem2}
Let $\Phi:X\to\mathbb{R}$ be defined as previously.
 Then $\Phi':X\to X^*$ satisfies $S_+$ condition
\end{lemma}

\begin{proof}
If $(u_n,v_n)\rightharpoonup (u,v)$ in $X$ and
\begin{equation}\label{eq.6}
\limsup_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v)
\rangle\leqslant 0
\end{equation}
then since $u_n\rightharpoonup u$ and
$v_n\rightharpoonup v$ in $W_0^{1,p}(\Omega)$ and
$W_0^{1,q}(\Omega)$ respectively
\begin{align*}
&\limsup_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v)\rangle\\
&=\limsup_{n\to\infty}(\int_\Omega
\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)(\nabla u_n-\nabla u)\,dx\\
&\quad + \int_\Omega \big(a_2(x,\nabla v_n)-a_2(x,\nabla v)\big)
(\nabla v_n-\nabla v)\,dx)
\end{align*}
and by \eqref{eq.3} and \eqref{eq.6},
\[
\lim_{n\to\infty}\langle\Phi'(u_n,v_n),(u_n-u,v_n-v)
\rangle=0,
\]
and obviously
\begin{gather}
\lim_{n\to\infty}\int_\Omega\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)
(\nabla u_n-\nabla u)\,dx=0,\label{eq.17}\\
\lim_{n\to\infty}\int_\Omega \big(a_2(x,\nabla v_n)-a_2(x,\nabla v)\big)
(\nabla v_n-\nabla v)\,dx=0.\label{eq.18}
\end{gather}
We shall prove $u_n\to u$ as a consequence of \eqref{eq.17},  and in a
similar way \eqref{eq.18} implies $v_n\to v$.
 By imitating the proof of  \cite[ Lemma 2.3]{DP}, put
 $$
P_n(x)=\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)\cdot(\nabla u_n-\nabla u).
$$
Then \eqref{eq.3} implies $P_n(x)\geqslant 0$ and because \eqref{eq.17},
there exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$
for which $\lim_{n\to\infty}P_n(x)=0$ a.e. in $\Omega$. Let
\[
E=\cap_{n\in\mathbb{N}}\{x\in\Omega : \lim_{n\to\infty}P_n(x)=0,\,
 |\nabla u_n(x)|<\infty,\, |\nabla u(x)|<\infty\}.
\]
Then $m(\Omega-E)=0$, $\lim_{n\to\infty}P_n(x)=0$ in $E$.

If $x_0\in E$ then by the Mean Value Theorem and  inequality \eqref{He},
\begin{align*}
&|\nabla u_n(x_0)|^p\\
&\leqslant C_2^{-1}A_1\big(x_0,\nabla u_n(x_0)\big)=C_2^{-1}
a_1\big(x_0,t_n\nabla u_n(x_0)\big)
\cdot\nabla u_n(x_0)\quad  \text{for some }t_n\in(0,1)\\
&\leqslant C_2^{-1}a_1\big(x_0,\nabla u_n(x_0)\big)
 \cdot\nabla u_n(x_0)\quad \text{by \eqref{eq.5}}\\
&\leqslant C_2^{-1}[P_n(x_0)+a_1(x_0,\nabla u_n(x_0))\nabla u(x_0)+
a_1(x_0,\nabla u(x_0))\cdot(\nabla u_n(x_0)-\nabla u(x_0))]\\
&\leqslant C_2^{-1}[P_n(x_0)+C_1(1+|\nabla u_n(x_0)|^{p-1})|
 \nabla u(x_0)|+C_1(1+|\nabla u(x_0)|^{p-1})|\nabla u_n(x_0)|\\
&\quad +a_1\big(x_0,\nabla u(x_0)\big)\cdot\nabla u(x_0)]\quad
 \text{by \eqref{Hb}}
\end{align*}
which implies $|\nabla u_n(x_0)|\leqslant C$ for some constant $C>0$.
 Because by our assumption $\lim_{n\to\infty}P_n(x_0)=0$, for any
 polynomial $q(t)=t^p+kt^{p-1}+mt+c$ with $p>1$,
\[
\lim_{t\to\infty}q(t)=\infty.
\]
Now, if $\nabla u_n(x_0)\nrightarrow\nabla u(x_0)$,
 then $\{\nabla u_n(x_0)\}$ has a convergent subsequence which is denoted
 by the same notation $\{\nabla u_n(x_0)\}$ and converges to a vector
$v_0\ne\nabla u(x_0)$. Hence
\[
\lim_{n\to\infty}P_n(x_0)= (a_1(x_0,v_0)-a_1\big(x_0,\nabla u(x_0)\big))
\cdot(v_0-\nabla u(x_0))>0,
\]
which contradicts the assumption $x_0\in E$. Therefore,
 $\nabla u_n(x)\rightarrow\nabla u(x)$ for every $x\in E$.

As a consequence, $P_n(x)\to 0$ a.e. in $\Omega$ and if
\[
g_n(x)=P_n(x)+\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)
\cdot\nabla u+a_1(x,\nabla u)\cdot(\nabla u_n-\nabla u)
+a_1(x,\nabla u)\cdot\nabla u
\]
then above calculations show that
\begin{equation}\label{eq.19}
|\nabla u_n(x)|^p\leqslant C_2^{-1}g_n(x);
\end{equation}
furthermore,
\begin{equation}\label{eq.20}
g_n(x)\to a_1(x,\nabla u)\cdot\nabla u
\end{equation}
a.e. in $\Omega$.
By Lemma \ref{lem1}, the hypothesis $(u_n,v_n)\rightharpoonup (u,v)$  implies
\begin{gather*}
\lim_{n\to\infty}\int_\Omega\big(a_1(x,\nabla u_n)-a_1(x,\nabla u)\big)
\cdot\nabla u\,dx
=\lim_{n\to\infty}\langle\Phi'(u_n,v_n)-\Phi'(u,v),(u,0)\rangle=0,
\\
\lim_{n\to\infty}\int_\Omega a_1(x,\nabla u)\cdot(\nabla u_n-\nabla u)\,dx
=\lim_{n\to\infty}\langle\Phi'(u,v),(u_n-u,0)\rangle=0.
\end{gather*}
On the other hand, \eqref{eq.17} gives
\[
\lim_{n\to\infty}\int_\Omega P_n(x)\,dx=0,
\]
and hence
\begin{equation}\label{eq.21}
\lim_{n\to\infty}\int_\Omega g_n(x)=\int_\Omega a_1(x,\nabla u)\cdot\nabla u.
\end{equation}
By \eqref{eq.19}, we obtain
\begin{align*}
|\nabla u_n(x)-\nabla u(x)|^p&\leqslant 2^{p-1}(|\nabla u_n(x)|^p
+|\nabla u(x)|^p)
\leqslant 2^{p-1}(C_2^{-1}g_n(x)+|\nabla u(x)|^p)
\end{align*}
and since $\nabla u_n(x)\to\nabla u(x)$ a.e. in $\Omega$, so \eqref{eq.20}
implies
\[
\lim_{n\to\infty}C_2^{-1}g_n(x)+|\nabla u(x)|^p
=C_2^{-1}a_1(x,\nabla u)\cdot\nabla u+|\nabla u(x)|^p,
\]
a.e. in $\Omega$. By \eqref{eq.21} we find
\begin{align*}
\lim_{n\to\infty}\int_\Omega C_2^{-1}g_n(x)+|\nabla u(x)|^p\,dx
&=\int_\Omega C_2^{-1}a_1(x,\nabla u)\cdot\nabla u+|\nabla u(x)|^p\,dx\\
&\leqslant C_2^{-1}\|a_1(x,\nabla u)\|_{L^{p'}(\Omega)}\|u\|_p+\|u\|_p^p.
\end{align*}
by the H\"{o}lder inequality in which $p'=\frac{p}{p-1}$.
Therefore, the Dominated Convergence Theorem implies
\[
\lim_{n\to\infty}\int_\Omega|\nabla u_n(x)-\nabla u(x)|^p\,dx=0,
\]
and therefore $u_n\to u$ in $W_0^{1,p}(\Omega)$.
Similarly we have $v_n\to v$ in $W_0^{1,q}(\Omega)$ and finally
$(u_n,v_n)\to (u,v)$ in $X$.
\end{proof}

\begin{lemma}\label{lem3}
 The functional $\Phi:X\to\mathbb{R}$ is weakly sequentially
lower semicontinuous and the functional $J:X\to\mathbb{R}$ is $C^1$
with compact derivative and $\Phi-\lambda J$ is weakly sequentially
lower semicontinuous and coercive for each
$\lambda\in\mathbb{R}$.
\end{lemma}

\begin{proof}
If $(u_n,v_n)\rightharpoonup (u,v)$ in $X$ and
$\liminf_{n\to\infty}\Phi(u_n,v_n)<\Phi(u,v)$ then there exists a subsequence
of $\{(u_n,v_n)\}$ denote it by  $\{(u_{n_k},v_{n_k})\}$ such that
 $\{\Phi(u_{n_k},v_{n_k})\}$ converges and
$\lim_{n\to\infty}\Phi(u_{n_k},v_{n_k})<\Phi(u,v)$.

Since $\Phi\in C^1(X;\mathbb{R})$ by Lemma \ref{lem1}, the Mean Value
Theorem implies the existence of $t_n\in (0,1)$ for every
$n\in\mathbb{N}$ such that
\[
\Phi(u_n,v_n)-\Phi(u,v)=\langle\Phi'\big(u+t_n(u_n-u),v+t_n(v_n-v)\big),
(u_n-u,v_n-v)\rangle.
\]
On the other hand, \eqref{eq.5} implies
\begin{equation}\label{eq.11}
\langle\Phi'(u,v),(\xi,\eta)\rangle\leqslant\langle\Phi'(u+t\xi,v+t\eta),
(\xi,\eta)\rangle
\end{equation}
for any $t\geqslant 0$ and $(\xi,\eta)\in X$. Therefore,
\[
\langle\Phi'(u,v),(u_n-u,v_n-v)\rangle
\leqslant\langle\Phi'\big(u+t_n(u_n-u),v+t_n(v_n-v)\big),(u_n-u,v_n-v)\rangle
\]
and as a consequence,
\begin{align*}
&\limsup_{k\to\infty}\langle\Phi'(u,v),(u_{n_k}-u,v_{n_k}-v)\rangle\\
&\leqslant\lim_{k\to\infty}\langle\Phi'\big(u+t_{n_k}(u_{n_k}-u),
 v+t_{n_k}(v_{n_k}-v)\big),(u_{n_k}-u,v_{n_k}-v)\rangle<0
\end{align*}
which contradicts $(u_n,v_n)\rightharpoonup (u,v)$ since
 $\Phi'(u,v)\in X^*$ by Lemma \ref{lem1}.
Thus $\liminf_{n\to\infty}\Phi(u_n,v_n)\geqslant\Phi(u,v)$ and
$\Phi:X\to\mathbb{R}$ is weakly sequentially lower semicontinuous.

It can be shown easily that $J$ is a $C^1$ functional
\cite[Theorem 2.9]{AP} and
\[
\langle J'(u,v),(\xi,\eta)\rangle=\int_\Omega g_1(x,u)\xi+g_2(x,v)\eta\,dx.
\]
If $\{(u_n,v_n)\}$ is a bounded sequence in $X$ then it has a weakly convergent
subsequence by reflexivity of $X$ which we also denote it by
$\{(u_n,v_n)\}$ and assume $(u_n,v_n)\rightharpoonup (u,v)$.
Since $1<p,q\leqslant n$, the embedding
$X\hookrightarrow L^p(\Omega)\times L^q(\Omega)$ is compact,
up to a subsequence $(u_n,v_n)\to (u,v)$ and by
\cite[Proposition 26.6]{Z}, the Nemytski operators
$g_1:L^p(\Omega)\to L^{p'}(\Omega)$ and $g_2:L^q(\Omega)\to L^{q'}(\Omega)$
are continuous and bounded where $p'=\frac{p}{p-1}$ and $q'=\frac{q}{q-1}$.
Then
\begin{align*}
&\big|\langle J'(u_n,v_n)-J'(u,v),(\xi,\mu)\rangle\big|\\
&\leqslant
\Big|\int_\Omega\big(g_1(x,u_n)-g_1(x,u)\big)\xi+\big(g_2(x,v_n)-g_2(x,v)\big)
 \mu\,dx\Big|\\
&\leqslant\|g_1(x,u_n)-g_1(x,u)\|_{L^{p'}(\Omega)}\|\xi\|_{L^p(\Omega)}
 +\|g_2(x,v_n)-g_2(x,v)\|_{L^{q'}(\Omega)}\|\mu\|_{L^q(\Omega)}\\
&\leqslant\max\big\{\|g_1(x,u_n)-g_1(x,u)\|_{L^{p'}(\Omega)},
 \|g_2(x,v_n)-g_2(x,v)\|_{L^{q'}(\Omega)}\big\}\|(\xi,\mu)\|
\end{align*}
hence
\begin{align*}
&\|J'(u_n,v_n)-J'(u,v)\|\\
&\leqslant\max\big\{\|g_1(x,u_n)-g_1(x,u)\|_{L^{p'}
(\Omega)},\|g_2(x,v_n)-g_2(x,v)\|_{L^{q'}(\Omega)}\big\},
\end{align*}
for any $n\in\mathbb{N}$ and $(\xi,\mu)\in X$. Therefore,
 $J':X\to X^*$ is compact and $J:X\to\mathbb{R}$ is weakly sequentially
continuous by Corollary 41.9 \cite{Z2}. Hence $\Phi-\lambda J$ are weakly
sequentially lower semicontinuous functionals on $X$ for every
$\lambda\in\mathbb{R}$.

By \eqref{He} we obtain
\begin{align*}
\Phi(u,v)&=\int_\Omega A_1(x,\nabla u)+A_2(x,\nabla v)\,dx\geqslant C_2\big(\|u\|_p^p+\|v\|_q^q\big)
\end{align*}
and since according to \eqref{eq.12},
\begin{equation}\label{eq.13}
\begin{aligned}
J(u,v)
&\leqslant \int_\Omega\Big|\int_0^ug_1(x,s)\,ds \Big|
 +\Big|\int_0^vg_2(x,s)\,ds\Big|\,dx\\
&\leqslant C  \int_\Omega(|u|+|u|^\tau+|v|+|v|^\kappa)\,dx\\
&\leqslant C(\|u\|_p^\tau+\|v\|_q^\kappa),
\end{aligned}
\end{equation}
we have
\[
\Phi(u,v)-\lambda J(u,v)
\geqslant C_2\big(\|u\|_p^p+\|v\|_q^q\big)-C|\lambda|
\big(\|u\|_p^\tau+\|v\|_q^\kappa\big).
\]
Then for every $\lambda\in\mathbb{R}$,
\[
\liminf_{\|(u,v)\|\to\infty}\Phi(u,v)-\lambda J(u,v)=\infty
\]
and hence $\mathcal{E}_\lambda=\Phi-\lambda J$ is coercive.
\end{proof}


Now we consider the properties of $\Psi$ that we need in this article.

\begin{lemma}\label{lem4}
Let $F:\overline\Omega\times\mathbb{R}^2\to\mathbb{R}$ be a Carath\'{e}dory 
function such that $F(x,0,0)\in L^1(\Omega)$ and $F(x,u,v)$ 
has continuous partial derivatives with respect to $u$ and $v$ in 
every $x\in\Omega$ and for some constant $C>0$
\[
|F_u(x,u,v)|\leqslant C(1+|u|^{p-1}+|v|^{q\frac{p-1}{p}}),\quad
|F_v(x,u,v)|\leqslant C(1+|u|^{p\frac{q-1}{q}}+|v|^{q-1})
\]
for every $x\in\Omega$ and $u,v\in\mathbb{R}$. Then
 $\Psi\in C^1(X;\mathbb{R})$ and its derivative $\Psi':X\to X^*$ is compact.
\end{lemma}

\begin{proof}
Since $F(x,u,v)$ is $C^1$ with respect to $u,v$, then for every $x\in\Omega$ 
there exist $\gamma(x),\theta(x)$ in $(0,1)$ such that
\begin{align*}
|F(x,u,v)-F(x,0,0)|&\leqslant |F(x,u,v)-F(x,u,0)|+|F(x,u,0)-F(x,0,0)|\\
&\leqslant |F_u(x,\gamma(x) u,0)||u|+|F_v(x,u,\theta(x) v)||v|\\
&\leqslant C(1+|u|^{p-1})|u|+C(1+|u|^{p\frac{q-1}{q}}+|v|^{q-1})|v|\\
&\leqslant C(1+|u|^p+|v|^q)
\end{align*}
hence $\Psi(u,v)\in\mathbb{R}$. Also for every $(u,v),(\xi,\mu)$ in $X$ 
and $t\in\mathbb{R}-\{0\}$, by the Mean Value Theorem,
\begin{align*}
& \lim_{t\to 0}\frac{\Psi(u+t\xi,v+t\mu)-\Psi(u,v)}{t}\\
&=\lim_{t\to 0}\frac{1}{t}\int_\Omega F\big(x,u(x)+t\xi(x),v(x)
 +t\mu(x)\big)-F\big(x,u(x),v(x)\big)\,dx\\
&=\lim_{t\to 0}\Big\{\int_\Omega F_u\big(x,u(x)
 +t\theta(x)\xi(x),v(x)+t\mu(x)\big)\xi(x)\,dx\\
&\quad+\int_\Omega F_v\Big(x,u(x),v(x)
 +t\gamma(x)\mu(x)\Big)\mu(x)\,dx\Big\},
\end{align*}
in which $0<\theta(x),\gamma(x)<1$ for any $x\in\Omega$. 
But $F_u$ is continuous and
\[
F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big)\to F_u\big(x,u(x),v(x)\big)
\quad\text{as }t\to 0
\]
and for $|t|<1$,
\begin{align*}
&\big|F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big)\xi(x)\big|\\
&\leqslant C\Big(1+(|u(x)|+|\xi(x)|)^{p-1}+(|v(x)|+|\mu(x)|)^{q\frac{p-1}{p}}\Big)|\xi(x)|
\end{align*}
therefore, the Dominated Convergence Theorem implies
\[
\lim_{t\to 0}\int_\Omega F_u\big(x,u(x)+t\theta(x)\xi(x),v(x)+t\mu(x)\big)
\xi(x)\,dx
= \int_\Omega F_u\big(x,u(x),v(x)\big)\xi(x)\,dx
\]
and similarly
\[
\lim_{t\to 0}\int_\Omega F_v\Big(x,u(x),v(x)+t\gamma(x)\mu(x)\Big)\mu(x)\,dx
= \int_\Omega F_v\Big(x,u(x),v(x)\Big)\mu(x)\,dx.
\]
Therefore,
\begin{align*}
 \langle\Psi'(u,v),(\xi,\mu)\rangle
&=\lim_{t\to 0} \frac{\Psi(u+t\xi,v+t\mu)-\Psi(u,v)}{t}\\
&=\int_\Omega F_u(x,u,v)\xi+F_v(x,u,v)\mu\,dx
\end{align*}
and $\Psi$ is G\^{a}teaux differentiable at any $(u,v)\in X$ and for every 
$(\xi,\mu)\in X$
\[
\langle\Psi'(u,v),(\xi,\mu)\rangle
=\int_\Omega F_u(x,u,v)\xi+F_v(x,u,v)\mu\,dx.
\]
The continuity and compactness of $\Psi'$ can be proved like the continuity 
of $\Phi'$ and the compactness of $J'$ respectively.
\end{proof}

Now we are ready to prove our next main result which deals with the existence 
of three weak solutions for \eqref{p}, by
introducing some controls on the behaviour of antiderivatives 
of $g_1$ and $g_2$ at zero.

\begin{theorem}\label{thm3}
 Let $g_1,g_2$ satisfy \eqref{eq.12} and suppose
\begin{equation}\label{eq.9}
 \max\big\{\limsup_{\xi\to 0}\frac{\sup_{x\in\Omega}G_1(x,\xi)}{|\xi|^p},\,
 \limsup_{\xi\to 0}\frac{\sup_{x\in\Omega}G_2(x,\xi)}{|\xi|^q}\big\}\leqslant 0,
\end{equation}
where
\[
G_1(x,\xi)=\int_0^{\hspace*{1pt}\xi} g_1(x,s)\,ds,\quad 
G_2(x,\xi)=\int_0^{\hspace*{1pt}\xi} g_2(x,s)\,ds
\]
for any $(x,\xi)\in\Omega\times\mathbb{R}$. Also, suppose the function 
$F:\overline\Omega\times\mathbb{R}^2\to\mathbb{R}$ satisfies all hypotheses 
of Lemma \ref{lem4} and in addition
\[
\sup\big\{J(u,v): (u,v)\in X\big\}>0.
\]
Then, if we set
\[
\gamma=\inf\big\{\frac{\Phi(u,v)}{J(u,v)}: (u,v)\in X,\,J(u,v)>0,
\,\Phi(u,v)>0\big\}
\]
for each compact interval $[a,b]\subset]\gamma,\infty[$ there exists $r>0$ 
such that for every $\lambda\in[a,b]$, there exists $\delta>0$ 
such that for every $\mu\in[0,\delta]$, the problem \eqref{p} has at 
least three weak solutions whose norms in $X$ are less than $r$.
\end{theorem}

\begin{proof}
First note that if $p\leqslant q$ then for every bounded $E\subset X$ 
there exists some constant $C>0$ such that
 \[
 \Phi(u,v)-\Phi(0,0)\geqslant C_2\big(\|u\|_p^p+\|v\|_q^q\big)
\geqslant C\big(\|u\|_p+\|v\|_q\big)^p=C\|(u,v)\|^p
 \]
 for every $(u,v)\in E$, and if $p>q$ then
 \[
\Phi(u,v)-\Phi(0,0)\geqslant C\|(u,v)\|^q.
\]
 Furthermore every weak solution of \eqref{p} is a solution of
 $\Phi'(x)=\lambda J'(x)+\mu \Psi'(x)$. Since $1<\tau<p,\ 1<\kappa<q$
\[
	\limsup_{\|(u,v)\|\to\infty}\frac{J(u,v)}{\Phi(u,v)}
\leqslant\limsup_{\|(u,v)\|\to\infty}\frac{C(\|u\|_p^\tau
+\|v\|_q^\kappa)}{\|u\|_p^p+\|v\|_q^q}=0
\]
and \eqref{eq.12} in conjunction with \eqref{eq.9} implies, there exist
$\rho_1,\rho_2$ so that $0<\rho_1<\rho_2$ and
 \[
 G_1(x,\xi)+G_2(x,\eta)\leqslant\epsilon(|\xi |^p+|\eta |^q)
 \]
for every $x\in\Omega$, every $\xi,\eta$ in
$\mathbb{R}-([-\rho_2,-\rho_1]\cup[\rho_1,\rho_2])$.
Since $G_1(x,\xi),\,G_2(x,\eta)$ are bounded on
$\Omega\times ([-\rho_2,-\rho_1]\cup[\rho_1,\rho_2])$, we can choose
$C'>0$ and $p<m<\frac{pn}{n-p}$ and $q<\ell<\frac{qn}{n-q}$ such that
 \[
G_1(x,\xi)+G_2(x,\eta)\leqslant\epsilon(|\xi |^p
+|\eta |^q)+C'(|\xi |^m+|\eta |^\ell)
\]
for all $(x,\xi)\in\Omega\times\mathbb{R}$. Now the continuity of the
Sobolev embedding implies for some constant $C$, independent of $\epsilon$
\[
J(u,v)\leqslant C\big(\epsilon(\|u\|_p^p+\|v\|_q^q)+C'(\|u\|_p^m
+\|v\|_q^\ell)\big)
\]
for every $(u,v)\in X$. On the other hand, \eqref{He} implies
$\Phi(u,v)\geqslant C_2(\|u\|_p^p+\|v\|_q^q)$ and since $p<m,\, q<\ell$
\begin{equation}\label{eq.14}
\limsup_{(u,v)\to(0,0)}\frac{J(u,v)}{\Phi(u,v)}\leqslant\frac{C\epsilon}{C_2}.
\end{equation}
Since $\epsilon>0$ is arbitrary
\[
\limsup_{(u,v)\to(0,0)}\frac{J(u,v)}{\Phi(u,v)}=0.
\]
Hence, by \eqref{eq.14} we have $\alpha=0$ in Theorem \ref{thm2} and
since all other hypotheses of Theorem \ref{thm2} for the functionals
$\Phi$ and $J$ and the point $x_0=(0,0)\in X$ are established
in Lemmas \ref{lem1}, \ref{lem2} and \ref{lem3} and the functional $\Psi$
 has needed properties by Lemma \ref{lem4}, therefore the result is proved.
\end{proof}


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\end{document}