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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 167, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/167\hfil Existence of solutions]
{Existence of solutions for non-uniformly nonlinear elliptic systems}

\author[G. A. Afrouzi, S. Mahdavi, N. B. Zographopoulos
\hfil EJDE-2011/167\hfilneg]
{Ghasem Alizadeh Afrouzi, Somayeh Mahdavi, \\
Nikolaos B. Zographopoulos} % in alphabetical order

\address{Ghasem Alizadeh Afrouzi \newline
Department of Mathematics, Faculty of Mathematical Sciences \\
University of Mazandaran, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}

\address{Somayeh Mahdavi \newline
Department of Mathematics, Faculty of Mathematical Sciences \\
University of Mazandaran, Babolsar, Iran}
\email{smahdavi@umz.ac.ir}

\address{Nikolaos B. Zographopoulos \newline
University of Military Education, Hellenic Army Academy \\
Department of Mathematics \& Engineering Sciences, 
 Vari - 16673, Athens, Greece}
\email{nzograp@gmail.com, zographopoulosn@sse.gr}

\thanks{Submitted November 12, 2011. Published December 14, 2011.}
\subjclass[2000]{34B18, 35B40, 35J65}
\keywords{Non-uniformly elliptic system; mountain pass theorem;
\hfill\break\indent  minimum principle}

\begin{abstract}
 Using a variational approach, we prove the existence of
 solutions for the degenerate quasilinear elliptic system
 \begin{gather*}
 -\operatorname{div}(\nu_1 (x)|\nabla u|^{p-2} \nabla u)
 =\lambda F_u(x,u,v)+\mu G_u(x,u,v),\\
 -\operatorname{div}(\nu_2 (x)|\nabla v|^{q-2} \nabla v)
 =\lambda F_v(x,u,v)+\mu  G_v(x,u,v),
 \end{gather*}
 with Dirichlet boundary conditions.
\end{abstract}

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

\section{Introduction}

In this article, we study the degenerate quasilinear elliptic
system
\begin{equation} \label{1}
\begin{gathered}
 -\operatorname{div}(\nu_1 (x)|\nabla u|^{p-2} \nabla u )
=\lambda F_u(x,u,v)+\mu   G_u(x,u,v), \quad \text{in }\Omega,\\
-\operatorname{div}(\nu_2 (x)|\nabla v|^{q-2} \nabla v)
=\lambda F_v(x,u,v)+\mu   G_v(x,u,v), \quad\text{in }\Omega,\\
 u=v=0, \quad\text{on } \partial\Omega.
\end{gathered}
\end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$,
$N \geq 2$ and  $1 < p,q < N$.
The parameters $\lambda$ and $\mu$ are nonnegative real numbers.

Throughout this work we assume that
\begin{equation} \label{eP}
(F_u,F_v)=\nabla F\quad\text{and}\quad (G_u,G_v)=\nabla G
\end{equation}
which stand for gradient of $F$ and $G$, respectively,
in the variables $w=(u,v) \in \mathbb{R}^2$.
Systems of form \eqref{1}, where
hypothesis \eqref{eP} is satisfied, are called potential systems. In
recent years, more and more attention have been paid to the
existence and multiplicity of positive solutions for potential
systems. For more details about this kind of systems see
\cite{ar73,bf02,ch96, ct09,dt03,dt07,sx08,sz99,st01,zt10,wz10}
 and references therein.

The degeneracy of this system is considered in
the sense that the measurable, non-negative diffusion
coefficients $\nu_1$, $\nu_2$ are allowed to vanish in
$\Omega$, (as well as at the boundary $\partial \Omega$) and/or
to blow up in $\bar{\Omega}$. The point of departure for the
consideration of suitable assumptions on the diffusion
coefficients is the work \cite{dkn97}, where the degenerate scalar
equation was studied.

We introduce the space
$(\mathcal{H})_p$  consisting of functions
$\nu: \Omega \subset \mathbb{R}^N \to \mathbb{R}$, such that
$\nu \in L^{1} (\Omega)$, $\nu^{-1/(p-1)} \in L^{1} (\Omega)$ and
$\nu^{-s} \in L^{1} (\Omega)$, for some $p>1$,
$s >\max\{\frac{N}{p}, \frac{1}{p-1}\}$ satisfying
$ps \leq N(s+1)$.

Then for the weight functions $\nu_1$, $\nu_2$ we assume the
 hypothesis:
\begin{itemize}
\item[(H1)]  There exist $\mu_1$ in the space
$(\mathcal{H})_p$ for some $s_p$,
and there exists $\mu_2$ in the spaces $(\mathcal{H})_{q}$
for some $s_p$, such that
\begin{equation} \label{e2.2}
\frac{\mu_1(x)}{c_1} \leq \nu_1(x) \leq c_1 \mu_1(x),\quad
\frac{\mu_2(x)}{c_2} \leq \nu_2(x) \leq c_2 \mu_2(x),
\end{equation}
a.e. in $\Omega$, for some constants $c_1 >1$ and $c_2>1$.
\end{itemize}

There exists a vast literature on non-uniformly
nonlinear elliptic problems in bounded or unbounded domains. Many
authors studied the existence of solutions for such problems
(equations or systems); see for example
\cite{ch08, cht09, mr09, tc09, zz09, z04,z08}.
Recently in \cite{cht09}, the authors considered the system
\begin{gather*}
  -\operatorname{div}(h_1 (x) \nabla u)
=\lambda F_u(x,u,v), \quad\text{in }\Omega,\\
 -\operatorname{div}(h_2 (x) \nabla v)
=\lambda F_v(x,u,v), \quad \text{in }\Omega,\\
 u=v=0, \quad\text{on } \partial\Omega.
\end{gather*}
They are concerned with the nonexistence and multiplicity of
nonnegative, nontrivial solutions.
In \cite{z08}, the author studied the principal eigenvalue of
the system
\begin{gather*} %1.3
 -\nabla (\nu_1(x)|\nabla u|^{p-2} \nabla u)
=\lambda   a(x)|u|^{p-2}u+\lambda b(x) |u|^{\alpha}|v|^{\beta}v,
 \quad \text{in }\Omega,\\
-\nabla (\nu_2(x)|\nabla v|^{q-2} \nabla v)
=\lambda d(x)|v|^{q-2}v+\lambda b(x) |u|^{\alpha}|v|^{\beta}u,
\quad\text{in }\Omega,\\
 u=v=0, \quad\text{on } \partial\Omega.
\end{gather*}
While in \cite{mr09} the following system was considered
\begin{gather*}
 -\operatorname{div}(|x|^{-ap}|\nabla u|^{p-2} \nabla u )
 =\lambda g_1(x,u,v), \quad \text{in }\Omega,\\
-\operatorname{div}(|x|^{-bq}|\nabla v|^{q-2} \nabla v)
 =\lambda g_2(x,u,v), \quad\text{in }\Omega,\\
u=v=0, \quad\text{on } \partial\Omega,
\end{gather*}
where $g_1, g_2:\Omega\times \mathbb{R}times\mathbb{R}$ are
continuous and monotone functions.

The aim of this work is to extend or complete some of the above
results for system \eqref{1}. Our assumptions are as follows:
$F(x,t,s)$ and $G(x,t,s)$ are $C^1$-functions satisfying the
hypotheses below:
\begin{itemize}
\item[(F1)] There exist positive constants $c_1, c_2>0$ such that
$$
|F_u(x,t,s)| \leq c_1|t|^{\theta}|s|^{\delta+1},\quad
|F_v(x,t,s)| \leq c_2|t|^{\theta+1}|s|^{\delta}
$$
for all $(t,s) \in \mathbb{R}^2$, a.e. $x \in \Omega$
and some $\theta, \delta >0$ with
\begin{equation} \label{1.4}
\frac{\theta+1}{p}+\frac{\delta+1}{q}=1.
\end{equation}

\item[(F2)]
\[
\lim_{|(s,t)| \to \infty} \frac{1}{p}F_u(x,s,t)
+\frac{1}{q}F_v(x,s,t)-F(x,s,t)=\infty
\]

\item[(G1)] There exist positive constants $c'_1, c'_2$
$$
G_u(x,t,s) \leq c'_1 |t|^\alpha|s|^{\gamma+1},\quad
G_v(x,t,s) \leq c'_2|t|^{\alpha+1}|s|^{\gamma};
$$
for all $(t,s) \in \mathbb{R}^2$, a.e. $x \in \Omega$ and
for some $\alpha, \gamma>0$. We will distinguish the
following  cases:
\begin{gather} \label{1.2}
\frac{\alpha+1}{p}+\frac{\gamma+1}{q}<1;\\
 \label{1.3}
\frac{\alpha+1}{p}+\frac{\gamma+1}{q}>1\quad
\textrm{and }\frac{\alpha+1}{p^*}+\frac{\gamma+1}{q^*}<1;
\end{gather}

\item[(G2)]
\[
 \lim_{|(s,t)| \to \infty} \frac{1}{p}G_u(x,s,t)
+\frac{1}{q}G_v(x,s,t)-G(x,s,t)=\infty
\]
\end{itemize}

The main results of this paper are the following two theorems.

\begin{theorem} \label{thm1}
In addition to {\rm (F1), (G1)} and \eqref{1.2}, assume that
there exist $p_1 \in (1,p)$ and $q_1 \in (1,q)$, such that
$\frac{\alpha+1}{p_1}+\frac{\gamma+1}{q_1}=1$.
Then there exists $\lambda_0>0$, such that \eqref{1} possesses
a weak solution for all $\mu>0$ and $0\leq \lambda<\lambda_0$.
\end{theorem}

\begin{theorem} \label{thm2}
In addition to {\rm (F1), (G1), (F2)} or {\rm (G2)} and \eqref{1.3},
assume that there exist $p_2 \in (p, p^*)$ and $ q_2 \in (q, q^*)$,
such that $\frac{\alpha+1}{p_2}+\frac{\gamma+1}{q_2}=1$. Then there
exists $\lambda_0>0$ such that system \eqref{1} possesses a weak
solution for all $\mu>0$ and $0\leq \lambda <\lambda_0$.
\end{theorem}

The quantities $p^*$ and $q^*$ are defined in the next section.

\section {Preliminaries}

Let $\nu(x)$ be a nonnegative weight function in $\Omega$
which satisfies condition $\mathcal{H}_p$. We consider the
weighted Sobolev space $\mathcal{D}^{1,p}_0 (\Omega, \nu)$
defined as the closure of $C_0^{\infty}(\Omega)$ with
respect to the norm
\[
\|u\|_{\mathcal{D}_0^{1,p}(\Omega,\nu)} := \Big(
\int_{\Omega} \nu(x) |\nabla u|^p  \Big)^{1/p}.
\]
The space $\mathcal{D}_0^{1,p}(\Omega,\nu)$ is a reflexive
Banach space. For a discussion about the space setting we refer
the reader to \cite{dkn97} and the references therein.
Let
\begin{equation} \label{e2.1}
p^{*}_{s} := \frac{Nps}{N(s+1)-ps}.
\end{equation}

\begin{lemma} \label{lemma2.1}
Assume that $\Omega$ is a bounded domain in $\mathbb{R}^N$ and
the weight $\nu$ satisfies $(\mathcal{N})_p$. Then the
following embeddings hold:
\begin{itemize}
\item[(i)]  $\mathcal{D}_0^{1,p}(\Omega,\nu) \hookrightarrow
L^{p^{*}_{s}}(\Omega)$ continuously for $1<p^{*}_{s}<N$,

\item[(ii)] $\mathcal{D}_0^{1,p}(\Omega,\nu) \hookrightarrow L^r
(\Omega)$ compactly for any $r \in [1,p^{*}_{s})$.
\end{itemize}
\end{lemma}

In the sequel we denote by $p^*$ and $q^*$ the quantities
$p^{*}_{s_p}$ and $p^{*}_{s_q}$, respectively, where $s_p$
and $s_q$ are induced by condition $(\mathcal{H})$, recall
that $\nu_1$, $\nu_2$ satisfy $(\mathcal{H})$.

The space setting for our problem is the product space
$H := \mathcal{D}_0^{1,p}(\Omega, \nu_1) \times
\mathcal{D}_0^{1,q}(\Omega, \nu_2)$ equipped with the norm
\[
\|h\|_H := \|u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)} +
\|v\|_{\mathcal{D}_0^{1,q}(\Omega, \nu_2)},\;\;\; h=(u,v) \in H.
\]
Observe that \eqref{e2.2} in condition $(\mathcal{H})$ implies
that the spaces $\mathcal{D}_0^{1,p}(\Omega, \nu_1) \times
\mathcal{D}_0^{1,q}(\Omega, \nu_2)$ and
$\mathcal{D}_0^{1,p}(\Omega, \mu_1) \times
\mathcal{D}_0^{1,q}(\Omega, \mu_2)$ are equivalent. Next, we
introduce the functionals
$I, J,\tilde{J} : H \to \mathbb{R}$ as follows:
\begin{gather*}
I(u,v) := \frac {1} {p} \int_{\Omega} \nu_1 (x) | \nabla
u|^{p}\, dx + \frac {1} {q} \int_{\Omega} \nu_2 (x) |
\nabla v|^{q}\, dx,  \\
J(u,v) := \int_{\Omega} F(x,u,v)\, dx, \\
\tilde{J}(u,v) := \int_{\Omega} G(x,u,v)\, dx.
\end{gather*}
It is a standard procedure (see \cite{dsz03, sz99}) to prove
the following properties of these functionals.

\begin{lemma} \label{l2.2}
The functionals $I, J, \tilde{J}$ are well
defined. Moreover, $I$ is continuous and $J, \tilde{J}$ are
compact.
\end{lemma}

We say that $(u,v)$ is a \emph{weak solution} of problem
\eqref{1} if $(u,v)$ is a critical point of the functional
$\Phi(u,v):=I(u,v) - \lambda J(u,v) - \mu \tilde{J}(u,v)$; i.e.,
\begin{gather}
\label{e2.3}
\int_{\Omega} \nu_1(x) |\nabla u|^{p-2}\nabla u \cdot \nabla \phi\, dx
 = \lambda \int_{\Omega} F_u (x,u,v)\phi\, dx + \mu
\int_{\Omega} G_u (x,u,v)\phi\, dx,\\
 \label{e2.4}
\int_{\Omega} \nu_2(x) |\nabla v|^{q-2}\nabla v \cdot \nabla \psi\, dx
=\lambda \int_{\Omega} F_v (x,u,v) \psi\, dx + \mu
\int_{\Omega} G_v (x,u,v) \psi\, dx,
\end{gather}
for any $(\phi,\psi) \in H$.

Also, we mention some results concerning the associated eigenvalue
problem. Let $\lambda_1$ be the first eigenvalue of the
Dirichlet problem
\begin{equation} \label{eigenproblem}
\begin{gathered}
-\operatorname{div}(\nu_1(x) |\nabla u|^{p-2}\nabla u)
=\lambda  |u|^{\theta-1}|v|^{\delta+1}u, \quad
 \text{in }\Omega,\\
 -\operatorname{div}(\nu_2 (x) |\nabla v|^{q-2}\nabla v)
=\lambda |u|^{\theta+1}|v|^{\delta-1} v, \quad
 \text{in }\Omega,\\
u=v=0,\quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
where the functions $\nu_1(x)$ and $\nu_2(x)$ satisfy (H1),
 and the exponents $\theta$, $\delta$ satisfy
\eqref{1.4}. Then, we have that $\lambda_1$ is a positive number,
which is characterized variationally by
\[
\lambda_1=\inf _{(u,v)\in H-\{(0,0)\}}
\frac{\int (\frac{\theta+1}{p} \nu_1(x) |\nabla u|^{p}
+ \frac{\delta+1}{q}\, \nu_2(x) |\nabla v|^{q})\, dx}
{\int |u|^{\theta+1} |v|^{\delta+1}\, dx}.
\]
Moreover, $\lambda_1$ is isolated, the associated eigenfunction
$(\varphi_1,\varphi_2)$ is componentwise nonnegative and $\lambda_1$
is the only eigenvalue of \eqref{eigenproblem} to which corresponds
a componentwise nonnegative eigenfunction. In addition, the set of
all eigenfunctions corresponding to the principal eigenvalue
$\lambda_1$ forms a one-dimensional manifold $E_1\subset H$, which
is defined by
$$
E_1=\{(t_1\varphi_1,t_1^{p/q}\varphi_2); t_1 \in \mathbb{R}\}.
$$
In the rest of this article, the following assumption is required.
\begin{equation}\label{2.5}
\lambda_1 \leq \liminf_{|(t,s)| \to \infty}\frac{\lambda
F(x,t,s)+\mu G(x,t,s)}{|t|^{\theta+1}|s|^{\delta+1}}
\end{equation}

\section{Proof of main theorems}

To prove Theorem \ref{thm1} we need following two Lemmas.

\begin{lemma} \label{l2.3}
Let $\{w_m\}$ be a sequence weakly converging  to $w$ in $H$.
Then we have
\begin{itemize}
\item[(i)] $\Phi(w) \leq \lim \inf_{m \to \infty} \Phi(w_m)$

\item[(ii)] $\lim_{m \to \infty} J(w_m)= J(w)$

\item[(iii)] $\lim_{m \to \infty} \tilde{J}(w_m)= \tilde{J}(w)$
\end{itemize}
\end{lemma}

\begin{proof} (i) Let $\{w_m\}=\{(u_m,v_m)\}$ be a sequence
that converges weakly to $w=(u,v) \in H$. By the weak lower
semicontinuity of the norm in the space
$\mathcal{D}_0^{1,p}(\Omega, \nu_1)$  and
$\mathcal{D}_0^{1,q}(\Omega, \nu_2)$, we deduce that
$$
\liminf_{m \to \infty} \int_{\Omega}
\nu_1(x) |\nabla u_m|^p   + \int_{\Omega} \nu_2(x) |\nabla
v_m|^q \geq    \int_{\Omega} \nu_1(x) |\nabla u|^p   +
\int_{\Omega} \nu_2(x) |\nabla v|^q.
$$
The compactness of operators
$J$ and $\tilde{J}$, by Lemma\eqref{l2.2}, imply the conclusion.
\end{proof}

\begin{lemma} \label{l3.2}
The functional $\phi$ is coercive and bounded from below.
\end{lemma}

\begin{proof}
 By (F1) and (G1), there exists $c_3, c'_3$, such that
for all $(t,s)\in \mathbb{R}^2$ and a. e. $x\in \Omega$,
we deduce that
$$ F(x,t,s)\leq c_3|t|^{\theta+1}|s|^{\delta+1},\quad
G(x,t,s)\leq c'_3|t|^{\alpha+1}|s|^{\gamma+1}.
$$
By taking $p_1\in (1,p$, $q_1 \in (1,q)$ such that
$\frac{\alpha+1}{p_1}+\frac{\gamma+1}{q_1}=1$ and applying Young's
inequality, we obtain
\begin{equation}\label{1.5}
\begin{split}
\int F(x,u,v) dx
&\leq c_3 \int|u|^{\theta+1}|v|^{\delta+1}dx\\
&\leq c_3(\frac{\theta+1}{p}\int|u|^p dx
  + \frac{\delta+1}{q}\int|v|^q dx )\\
& \leq c_3(\frac{\theta+1}{p}s_1\int \nu_1(x)|\nabla u|^p dx
 + \frac{\delta+1}{q}s_2\int \nu_2(x)|\nabla v|^q dx)\\
&\leq c(\frac{\theta+1}{p}\|u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)}^p
 + \frac{\delta+1}{q}\|v\|_{\mathcal{D}_0^{1,q}(\Omega, \nu_2)}^q)
\end{split}
\end{equation}
where $s_1, s_2$ are the embedding constants of
$\mathcal{D}_0^{1,p}(\Omega, \nu_1)\hookrightarrow L^p(\Omega)$,
$\mathcal{D}_0^{1,q}(\Omega, \nu_2)\hookrightarrow L^q(\Omega)$
and $c=\max\{c_3s_1, c_3s_2\}$, while
\begin{equation}\label{1.6}
\begin{split}
\int G(x,u,v) dx
&\leq c'_3\int|u|^{\alpha+1}|v|^{\gamma+1}dx\\
&\leq c'_3\frac{\alpha+1}{p_1}\int|u|^{p_1}dx
 +c'_3\frac{\gamma+1}{q_1}\int|v|^{q_1}dx\\
&\leq c'(\frac{\alpha+1}{p_1}\|u\|_{\mathcal{D}_0^{1,p}(\Omega,
 \nu_1)}^{p_1}
 +\frac{\gamma+1}{q_1}\|v\|_{\mathcal{D}_0^{1,q}(\Omega,\nu_2)}^{q_1})
\end{split}
\end{equation}
Consequently, using \eqref{1.5}, \eqref{1.6}, we obtain the
estimate
\begin{align*}
\Phi(u,v)
&\geq (\frac{1}{p}-\lambda c\frac{\theta+1}{p})
 \|u\|_{\mathcal{D}_0^{1,p}(\Omega,\nu_1)}^p
 +(\frac{1}{q}-\lambda c\frac{\delta+1}{q})
 \|v\|_{\mathcal{D}_0^{1,q}(\Omega, \nu_2)}^{q}\\
&\quad -\mu c'\frac{\alpha+1}{p_1}\|u\|_{\mathcal{D}_0^{1,p}
 (\Omega,\nu_1)}^{p_1}-\mu c'\frac{\gamma+1}{q_1}
 \|v\|_{\mathcal{D}_0^{1,q}(\Omega,\nu_2)}^{q_1}.
\end{align*}
Taking $\lambda_0>0$ such that
$\min \{1-\lambda(\theta+1) c,1-\lambda(\delta+1) c\}>0$ for all
$0 \leq \lambda <\lambda_0$, it follows that for $\mu >0$ and
$0 \leq \lambda <\lambda_0$, $\phi$ is coercive, indeed
$\phi(u,v) \to \infty$ as $\|(u,v)\|_H \to \infty$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
The coerciveness of $\Phi$ and the weak sequential lower
semicontinuity are enough in order to prove that $\Phi$ attains its
infimum, so the system \eqref{1} has at least one weak
solution.
\end{proof}

\begin{proof}[Proof of theorem \ref{thm2}]
 To prove the existence of a weak solution
we apply a version of the Mountain Pass Theorem due to Ambrosetti
and Rabinowitz [1]. For this purpose we verify that
$\Phi$ satisfies:
\begin{itemize}
\item[(i)] the mountain pass type geometry,

\item[(ii)] the $(PS)_c$ condition.
\end{itemize}

(i) By choosing $p_2 \in (p,p^*)$ and  $ q_2 \in (q,q^*)$
such that $\frac{\alpha+1}{p_2}+\frac{\gamma+1}{q_2}=1$ and applying
the Young's inequality, we obtain
\begin{align*}
\int G(x,u,v) dx
&\leq c_3'\int|u|^{\alpha+1}|v|^{\gamma+1}dx\\
&\leq c_3'(\frac{\alpha+1}{p_2}\int|u|^{p_2}dx
 +\frac{\gamma+1}{q_2}\int|v|^{q_1} dx) \\
&\leq c(\frac{\alpha+1}{p_2}\|u\|_{\mathcal{D}_0^{1,p}(\Omega,
 \nu_1)}^{p_2}
+\frac{\gamma+1}{q_2}\|v\|_{\mathcal{D}_0^{1,q}(\Omega,\nu_2)}^{q_2}),
\end{align*}
which implies
\begin{align*}
\Phi(u,v) &\geq (\frac{1}{p}-\lambda c\frac{\theta+1}{p})
 \|u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)}^p
 +(\frac{1}{q}-\lambda c\frac{\delta+1}{q})
 \|v\|_{\mathcal{D}_0^{1,q}(\Omega, \nu_2)}^q \\
&\quad -\mu c'\frac{\alpha+1}{p_2}\|u\|_{\mathcal{D}_0^{1,p}
 (\Omega,\nu_1)}^{p_2}-\mu c'\frac{\gamma+1}{q_2}
 \|v\|_{\mathcal{D}_0^{1,q}(\Omega, \nu_2)}^{q_2}.
\end{align*}
Hence, there exists $r>0$, small enough, such that
$$
\inf_{\|(u,v)\|=r}\Phi(u,v)>0=\Phi(0,0).
$$
On the other hand by using \eqref{2.5} we have
\begin{align*}
&\Phi(t^{1/p}\varphi_1, t^{1/q}\varphi_2)\\
&\leq \frac{t}{p}\int \nu_1|\nabla \varphi_1|^p dx
 + \frac{t}{q}\int \nu_2|\nabla \varphi_2|^q dx
 -(\lambda_1+\varepsilon)
 \int (|t^{1/p}\varphi_1|^{\theta+1}
 |t^\frac{1}{q}\varphi_2|^{\delta+1})dx \\
&=-t\varepsilon \int (|\varphi_1|^{\theta+1}|\varphi_2|^{\delta+1}) dx.
\end{align*}
Thus, we conclude that there exists $t>0$, large enough, such that for
$e=(t^{1/p}\varphi_1, t^{1/q}\varphi_2)$, we have
$\|e\|>r$ and $\Phi(e)<0$.

(ii) Let $\{w_n\}_{n=1}^{\infty} \in H$ be such that there
exists $c>0$, with
\begin{equation}\label{1.7}
|\Phi(w_n)| \leq c, \quad \forall n \in \mathbb{N},
\end{equation}
and there exists a strictly decreasing sequence
$\{\varepsilon_n\}_{n=1}^{\infty}, \lim_{n \to \infty}
\varepsilon_n=0$, such that
\begin{equation}\label{1.8}
|\langle\Phi'(w_n),z\rangle| \leq \varepsilon_n\|z\|_H,
\quad \forall n \in N , z \in H.
\end{equation}

We will prove that $\{w_n\}$ contains a subsequence which
converges strongly in $H$.
Let us begin by proving that $\{w_n\}$ is bounded in $H$.
Suppose, by contradiction, that $\|w_n\|_H \to \infty$. We have
\begin{align*}
&|\langle\Phi'(u_n,v_n),(u_n,v_n)\rangle| \\
&= | \int \nu_1(x)|\nabla u_n|^p dx
 +\int \nu_2(x)|\nabla v_n|^q dx -\lambda\int F_u(x,u_n,v_n)u_n dx\\
&\quad -\lambda \int F_v(x,u_n,v_n)v_n dx
 -\mu\int G_u(x,u_n,v_n)u_n dx
 -\mu \int G_v(x,u_n,v_n)v_n dx | \\
&\leq \varepsilon_n\|(u_n,v_n)\|_{H}.
\end{align*}
On the other hand
\begin{align*}
|\Phi(u_n,v_n)|
&= |\frac{1}{p}\int \nu_1(x)|\nabla u_n|^p dx
 +\frac{1}{q}\int \nu_2(x)|\nabla v_n|^q dx \\
&\quad -\lambda \int F(x,u_n,v_n) dx-\mu\int G(x,u_n,v_n) dx|
\leq c.
\end{align*}
Thus one has
\begin{align*}
&c+\varepsilon_n\|(u_n,v_n)\|_H \\
&\geq \Phi(u_n,v_n)-\langle \Phi'(u_n,v_n)
,(\frac{u_n}{p},\frac{v_n}{q})\rangle \\
&= \lambda \int\Big(\frac{1}{p}F_u(x,u_n,v_n)u_n
 +\frac{1}{q}F_v(x,u_n,v_n)v_n-F(x,u_n,v_n)\Big)  dx \\
&\quad \mu \int\Big(\frac{1}{p}G_u(x,u_n,v_n)u_n
 +\frac{1}{q}G_v(x,u_n,v_n)v_n-G(x,u_n,v_n)\Big)dx ,
\end{align*}
which contradicts both (F2) and (G2).
So $\{w_n\}$ is bounded.
This imply that there exists $(u,v) \in H$ such that at least
its subsequence, $w_n$ converges  and strongly
in $L^p(\Omega)\times L^q(\Omega)$. Choosing $z=(u_n-u,0)$ in
\eqref{1.8}, we obtain
\begin{align*}
&\Big|\int \nu_1(x)|\nabla u_n|^{p-2}\nabla u_n\nabla (u_n-u) dx
 -\lambda\int F_u(x,u_n,v_n)(u_n-u) dx \\
& -\mu\int G_u(x,u_n,v_n)(u_n-u) dx\Big|\\
& \leq \varepsilon_n\|u_n-u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)},
\end{align*}
\begin{align*}
\Big|\int F_u(x,u_n,v_n)(u_n-u) dx\Big|
&\leq \int |F_u(x,u_n,v_n)\|(u_n-u)| dx\\
&\leq \int |u_n|^{\theta}|v_n|^{\gamma+1}|u_n-u| dx \\
&\leq \|u_n\|_{L^p}^{\theta}\|v_n\|_{L^q}^{\gamma+1}\|u_n-u\|_{L^p},
\end{align*}
and
\begin{align*}
\Big|\int G_u(x,u_n,v_n)(u_n-u) dx\Big|
&\leq \int |G_u(x,u_n,v_n)\|(u_n-u)| dx\\
&\leq \int |u_n|^{\alpha}|v_n|^{\delta+1}|u_n-u| dx \\
&\leq \|u_n\|_{L^p}^{\alpha}\|v_n\|_{L^q}^{\delta+1}\|u_n-u\|_{L^p}.
\end{align*}
Thus, we obtain
$$
\int \nu_1(x)|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n- \nabla u)dx \to 0,
$$
as $n \to \infty$.
%
In the same way we obtain
$$
\int \nu_1(x)|\nabla u|^{p-2}\nabla u(\nabla u_n- \nabla u) dx,
$$
as $n \to \infty$. Finally, we conclude that
\begin{equation} \label{1.9}
\lim _{n \to \infty}\int \nu_1(x)(|\nabla
u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u) (\nabla u_n - \nabla
u) dx=0.
\end{equation}
Observe now that for all $\xi, \eta \in \mathbb{R}^N$,
there exists constant $c_3>0$, such that
\begin{equation}\label{1.10}
\begin{gathered}
(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta)
 \geq c(|\xi|+|\eta|)^{p-2}|\xi-\eta|^2  \quad\text{if }1<p<2\\
(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta,\xi-\eta)\geq c|\xi-\eta|^p
\quad\text{if } p \geq 2.
\end{gathered}
\end{equation}
where $(\cdot,\cdot)$ denotes the usual product in
$\mathbb{R}^N$.

So, for $1<p<2$, by H\"{o}lder's inequality and substituting
$z_n=\nu_1^{1/p}u_n, z=\nu_1^{1/p}u$ in \eqref{1.10},
there exists $c^*>0$, such that
\begin{align*}
0 &\leq  \int |\nabla z_n-\nabla z|^p dx\\
&=\int |\nabla z_n-\nabla z|^p (|\nabla z_n|+|\nabla z|)
 ^{p(p-2)/2}(|\nabla z_n|+|\nabla z|)^{p(2-p)/2} dx\\
&\leq  \Big(\int |\nabla z_n-\nabla z|^2
 (|\nabla z_n|+|\nabla z|)^{p-2}dx \Big)^{p/2}
 \Big(\int(|\nabla z_n|+|\nabla z|)^pdx\Big)^{(2-p)/2} \\
&\leq \frac{1}{c^*}\Big(\int (|\nabla z_n|^{p-2}\nabla z_n-|\nabla
 z|^{p-2}\nabla z, (\nabla z_n - \nabla z)dx\Big)^{p/2}\\
&\quad\times \Big(\int(|\nabla z_n|+|\nabla z|)^pdx\Big)^{(2-p)/2}\\
&\leq  \frac{c}{c^*}\Big(\int (|\nabla z_n|^{p-2}\nabla z_n
 -|\nabla z|^{p-2}\nabla z, (\nabla z_n
 - \nabla z)dx\Big)^{p/2},
\end{align*}
which implies
$\|u_n-u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)} \to 0$,
by \eqref{1.9}, as $n \to \infty$. While, for
$p\geq 2$, by \eqref{1.10}, one has
$$
0 \leq \|u_n-u\|_{\mathcal{D}_0^{1,p}(\Omega,
\nu_1)}\leq \frac{1}{c^*}\Big(\int (|\nabla z_n|^{p-2}\nabla
z_n-|\nabla z|^{p-2}\nabla z, (\nabla z_n - \nabla z)dx\Big),
$$
so we have $\|u_n-u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)} \to 0$,
by \eqref{1.9}, as $n \to \infty$. Therefore,
$\|u_n-u\|_{\mathcal{D}_0^{1,p}(\Omega, \nu_1)} \to 0$ for $p>1$,
as $n \to \infty$, that is, $u_n \to u$ in
${\mathcal{D}_0^{1,p}(\Omega, \nu_1)}$ as $n \to \infty$.
Similarly , we obtain $v_n \to v$ in
${\mathcal{D}_0^{1,q}(\Omega, \nu_2)}$ as $n \to \infty$.
Consequently, $\Phi$ satisfies the $(PS)_c$ condition and the proof
of  is completed.
\end{proof}

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\end{document}