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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 163, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/163\hfil $(p,q)$-Laplacian elliptic systems]
{$(p,q)$-Laplacian elliptic systems at resonance}

\author[Z.-Q. Ou \hfil EJDE-2016/163\hfilneg]
{Zeng-Qi Ou}

\address{Zeng-Qi Ou \newline
School of Mathematics and Statistics,
Southwest University, Chongqing 400715, China}
\email{ouzengq707@sina.com, Phone +86 23 68253135}


\thanks{Submitted June 25, 2015. Published June 28, 2016.}
\subjclass[2010]{35D30, 35J50, 35J92}
\keywords{Elliptic systems; Landesman-Lazer-type conditions; resonance; 
\hfill\break\indent critical point theorem}

\begin{abstract}
 We show the existence of weak solutions for a class of $(p,q)$-Laplacian elliptic
 systems at resonance, under certain Landesman-Lazer-type conditions by
 using critical point theorem.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and statement of main results}

   Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$
and $\Delta_p$ be the $p$-Laplacian operator. In this paper, we
study the existence of solutions for the problem
\begin{equation} \label{P1} 
\begin{gathered}
-\Delta_pu=\lambda_1|u|^{p-2}u+\frac{\lambda_1}{\beta+1}
|u|^{\alpha}|v|^{\beta}v+G_s(x,u,v)-h_1(x) \quad\text{in } \Omega ,\\
-\Delta_qv=\lambda_1 |v|^{q-2}v+\frac{\lambda_1}{\alpha+1}
|u|^{\alpha}|v|^{\beta}u+G_t(x,u,v)-h_2(x) \quad\text{in } \Omega ,\\
u=v=0 \quad\text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $1< p,q<+\infty$ and $\alpha\geq0,\beta\geq0$ satisfy
\begin{equation}\label{P2}
\frac{\alpha+1}{p}+\frac{\beta+1}{q}=1.
\end{equation}
The nonlinearity $G:\Omega\times \mathbb{R}^2\to R$ is a Caratheodory
function which has continuous derivatives $G_s(x,s,t),G_t(x,s,t)$
with respect to $s$ and $t$ for almost any $x\in \Omega$, and
$h_1\in L^{p/(p-1)}(\Omega), h_2\in L^{q/(q-1)}(\Omega)$.

  Let $W=W^{1,p}_0(\Omega)\times W^{1,q}_0(\Omega)$ with the
norm $\|(u,v)\|=\|u\|_p+\|v\|_q$ for all $(u,v)\in W$, where
$W^{1,p}_0(\Omega)$ is the usual Banach space with the norm
$\|u\|_p=\left(\int_\Omega|\nabla u|^pdx \right)^{1/p}$ for any
$u\in W^{1,p}_0(\Omega)$. From Sobolev embedding Theorem, the embedding
$W^{1,p}_0(\Omega)\hookrightarrow L^p(\Omega)$ is continuous and compact, and
there is constant $C>0$ such that
\begin{equation}\label{P3}
\|u\|_{L^p}\leq C\|u\|_p, \; \forall u\in W^{1,p}_0(\Omega),
\quad \text{and}\quad \|v\|_{L^q}\leq C\|v\|_q, \; \forall v\in W^{1,q}_0(\Omega),
\end{equation}
where $\|\cdot\|_{L^p}$ denotes the norm of $L^p(\Omega)$ and
throughout this paper, let $C$ always denote an embedding constant
with relation to \eqref{P3}. For the following nonlinear eigenvalue
problem
\begin{equation}\label{P4}
\begin{gathered}
-\Delta_pu=\lambda |u|^{p-2}u+\frac{\lambda}{\beta+1}
|u|^{\alpha}|v|^{\beta}v \quad\text{in } \Omega ,\\
-\Delta_qv=\lambda |v|^{q-2}v+\frac{\lambda}{\alpha+1}
|u|^{\alpha}|v|^{\beta}u \quad\text{in } \Omega ,\\
u=v=0 \quad\text{on } \partial \Omega,
\end{gathered}
\end{equation}
consider the functionals $\phi,\varphi$ on $W$ defined by
\begin{gather*}
\phi(u,v)=\frac{1}{p}\int_\Omega|\nabla u|^pdx
+\frac{1}{q}\int_\Omega|\nabla v|^qdx, \\
\varphi(u,v)=\frac{1}{p}\int_\Omega |u|^pdx
+\frac{1}{q}\int_\Omega|v|^qdx+\frac{1}{(\alpha+1)(\beta+1) }\int_{\Omega}|u|^{\alpha}|v|^{\beta}uvdx,
\end{gather*}
and the manifold
$$
\Sigma=\{(u,v)\in W:\varphi(u,v)=1\}.
$$
It is easy to prove that $\phi(u,v),\varphi(u,v)$ are
$(p,q)$-homogeneous, namely
\[
\phi(t^{1/p}u,t^{1/q}v)=t\phi(u,v),\quad
\varphi(t^{1/p}u,t^{1/q}v)=t\varphi(u,v)
\]
for any $ t>0$  and $(u,v)\in W$,
and $\Sigma$ is a symmetric nonempty manifold in $W$. By an argument similar to the ones in \cite{DR,OT}, problem \eqref{P4} has a sequence of eigenvalues with the variational characterization
$$
\lambda_k=\inf_{\Lambda\in\Sigma_k}\sup_{(u,v)\in \Lambda}\phi(u,v),
$$
where $\Sigma_k=\{\Lambda\subset\Sigma:\text{ there is an odd, continuous
and surjective }\gamma:S^{k-1}\to \Lambda\}$ and $S^{k-1}$ denotes the
unit sphere in $\mathbb{R}^k$.

On the other hand, let
$$
\lambda_1'=\inf_{(u,v)\in \Sigma}\phi(u,v),
$$
we can see that $\lambda_1=\lambda'_1$. Moreover, $\lambda_1$ is a simple, 
isolated and positive principal eigenvalue of \eqref{P4} and has a positive
 normalized eigenvalue $(\mu_0,\nu_0)$, namely, $\|\mu_0\|_p+\|\nu_0\|_q=1$.
 By a simple computation, there exists a positive constant $t_0$ such that
$$
\|t_0^{1/p}\mu_0\|_p^p+\|t_0^{1/q}\nu_0\|_q^q=1.
$$
Let $\mu_1=t_0^{1/p}\mu_0,\nu_1=t_0^{1/q}\nu_0$, since $\phi,\varphi$ are 
$(p,q)$-homogeneous, hence the set of all eigenfunctions corresponding 
to $\lambda_1$ is
$$
E_1:=\{(t^{1/p}\mu_1,t^{1/q}\nu_1):t\geq0\}
\cup\{(-t^{1/p}\mu_1,-t^{1/q}\nu_1):t\geq0\}.
$$
The set $E_1$ is not an one-dimensional linear subspace of $W$ and the 
corresponding orthogonal decomposition on $W$ does not hold with respect 
to the the first eigenvalue $\lambda_1$.

 In many papers, existence of weak solutions for the resonant elliptic problems 
were investigated under the well-known Landesman-Lazer-type conditions, 
which were introduced
by Landesman and Lazer in \cite{LL} and were extended by Tang in \cite{T1}. 
Since then they were used widely for the different types of equations, for example, 
in \cite{AO, DR, ST} for the quasilinear elliptic equations, in \cite{KT} 
for asymptotically linear noncooperative elliptic systems, in \cite{T2} for 
the forced duffing equations, in \cite{Sun} for Kirchhoff type equations. Especially, in \cite{S} the case $p=q=2$(the semilinear elliptic systems) was considered and the case $p=q\geq2$(the quasilinear elliptic systems) was discussed in \cite{OL,OT,Z} where $G_s(x,s,t)=g_1(x,s)$
and $G_t(x,s,t)=g_2(x,t)$. As far as we know, when $p\neq q>1$, the similar 
results are not discussed under the Landesman-Lazer-type conditions due 
to Landesman and Lazer. Motivated by these finding, we consider the existence 
of solutions for problem \eqref{P1} at resonance with the first eigenvalue
under the Landesman-Lazer-type conditions.
We first state the following fundamental hypotheses.
\begin{itemize}
\item[(H1)] There is $h\in C(\bar{\Omega}, \mathbb{R}^+)$ such that
$|G_s(x,s,t)|\leq h(x)$  and $ |G_t(x,s,t)|\leq h(x)$ 
for all $(x,s,t)\in \Omega\times \mathbb{R}^2$.

\item[(H2)] There exist two functions $g_1^{++},g_1^{--}\in C(\Omega,R)$ such that
\[
g_1^{++}(x)=\liminf_{s\to+\infty,\, t\to+\infty }G_s(x,s,t),\quad
g_1^{--}(x)=\limsup_{s\to-\infty,\, t\to-\infty}G_s(x,s,t)
\]
 uniformly a.e. $x\in \Omega$.

\item[(H3)] There is two functions $g_2^{++},g_2^{--}\in C(\Omega,
R)$ such that
\[
g_2^{++}(x)=\liminf_{ s\to+\infty,\, t\to+\infty}G_t(x,s,t),\quad
g_2^{--}(x)=\limsup_{ s\to-\infty,\, t\to-\infty}G_t(x,s,t)
\]
uniformly a.e. $x\in \Omega$.

\end{itemize}
The Landesman-Lazer-type conditions for problem \eqref{P1} are read either
\begin{equation}\label{P5}
\begin{aligned}
\int_\Omega g_1^{--}\mu_1dx+\int_\Omega g_2^{--}\nu_1dx
&<\int_\Omega h_1\mu_1dx+ \int_\Omega h_2\nu_1dx\\
&<\int_\Omega g_1^{++}\mu_1dx+\int_\Omega g_2^{++}\nu_1dx;
\end{aligned}
\end{equation}
or
\begin{equation}\label{P6}
\begin{aligned}
\int_\Omega g_1^{++}\mu_1dx+\int_\Omega g_2^{++}\nu_1dx
&<\int_\Omega h_1\mu_1dx+\int_\Omega h_2\nu_1dx \\
&<\int_\Omega g_1^{--}\mu_1dx+\int_\Omega g_2^{--}\nu_1dx.
\end{aligned}
\end{equation}
We are ready to state the main results.

\begin{theorem} \label{thm1}
 Let $h_1\in L^{p/(p-1)}(\Omega)$, $h_2\in L^{q/(q-1)}(\Omega)$, and
\eqref{P2}, {\rm (H1), (H2), (H3)} and \eqref{P5} be satisfied. 
If $1<p<q$ and the following inequalities hold:
\begin{equation}\label{P7}
\int_\Omega h_1\mu_1dx-\int_\Omega g^{++}_1\mu_1dx<0,\quad 
\int_\Omega h_1\mu_1dx-\int_\Omega g_1^{--}\mu_1dx>0,
\end{equation} 
then problem \eqref{P1} has at least one solution. 
\end{theorem}

In the other case $1<q<p$, the following result holds.

  \begin{theorem} \label{thm2}
 Let $h_1\in L^{p/(p-1)}(\Omega)$, $h_2\in L^{q/(q-1)}(\Omega)$, and
\eqref{P2}, {\rm (H1), (H2), (H3)} and \eqref{P5} be satisfied. 
If $1<q<p$ and the following inequalities hold:
\begin{equation}\label{P8}
\int_\Omega h_2\nu_1dx-\int_\Omega g^{++}_2\nu_1dx<0,\quad 
\int_\Omega h_2\nu_1dx-\int_\Omega g_2^{--}\nu_1dx>0,
\end{equation}
then problem \eqref{P1} has at least one solution.
\end{theorem}

\begin{theorem} \label{thm3}
 Let $h_1\in L^{p/(p-1)}(\Omega), h_2\in L^{q/(q-1)}(\Omega)$. 
If \eqref{P2}, {\rm (H1), (H2), (H3)} and \eqref{P6} are satisfied, then problem
\eqref{P1} has at least one solution.
\end{theorem}


Our results extends the ones in \cite{S}
from the semilinear elliptic systems to $(p,q)$-Laplacian elliptic systems,
and are also the generalizations of \cite{Z}, where they considered the case
$p=q\geq2$ and $G_s(x,s,t)=g_1(s), G_t(x,s,t)=g_2(t)$. Moreover, the
conditions \eqref{P7} and \eqref{P8} are the technical assumptions.
Theorem \ref{thm2} is similar to Theorem \ref{thm1}, and we will prove 
Theorem \ref{thm1} and Theorem \ref{thm3}.

\section{Proofs of Theorems}

Now consider the functionals $J, J_1, J_2$ on $W$ defined by
\begin{align*}%\label{E4}
J(u,v)&= \frac{1}{p}\int_\Omega|\nabla u|^pdx
+\frac{1}{q}\int_\Omega|\nabla
v|^qdx-\frac{\lambda_1}{p}\int_\Omega|u|^pdx\\
&\quad -\frac{\lambda_1}{q}\int_\Omega |v|^qdx-
\frac{\lambda_1}{(\alpha+1)(\beta+1)}\int_{\Omega}|u|^{\alpha}|v|^{\beta}uvdx\\
&\quad -\int_\Omega G(x,u,v)dx+\int_\Omega h_1udx+\int_\Omega h_2vdx,
\end{align*}
\begin{align*}
J_1(u,v)&= \frac{1}{p}\int_\Omega|\nabla u|^pdx
-\frac{\lambda_1}{p}\int_\Omega|u|^pdx
-\frac{\lambda_1}{p(\beta+1)}\int
_{\Omega}|u|^{\alpha}|v|^{\beta}uvdx\\
&\quad -\int_\Omega\int_0^1G_s(x,ru,rv)u\,dr\,dx+\int_\Omega h_1udx,
\end{align*}
\begin{align*}
J_2(u,v)&= \frac{1}{q}\int_\Omega|\nabla v|^qdx
-\frac{\lambda_1}{q}\int_\Omega|v|^qdx
-\frac{\lambda_1}{q(\alpha+1)}\int
_{\Omega}|u|^{\alpha}|v|^{\beta}uvdx\\
&\quad -\int_\Omega\int_0^1G_t(x,ru,rv)v\,dr\,dx+\int_\Omega h_2vdx.
\end{align*}
Noting that
\begin{equation}\label{P9}
G(x,s,t)=\int_0^1G_s(x,rs,rt)sdr+\int_0^1G_t(x,rs,rt)tdr,
\end{equation}
from \eqref{P2} and \eqref{P9}, it follows that
$$
J(u,v)=J_1(u,v)+J_2(u,v) \quad  \text{for all }(u,v)\in W.
$$
From (H1), it is easy to prove that the functional $J$ is well defined and 
$J\in C^1(W,R)$. Moreover, from the variational view of point, a weak
 solution of problem \eqref{P1} is
equivalent to a critical point of the functional $J$ in $W$. 
In this paper, we will prove Theorem \ref{thm1} and Theorem \ref{thm2}
 by using the  following G-linking Theorem due to Dr\'abek and Robinson 
(see \cite{DR,ST}) and Theorem \ref{thm3} by using Ekeland's Variational Principle
(see \cite{R,S2}). In these abstract theorems, a compact condition, i.e., 
$(PS)$ condition, is needed.

\begin{definition} \label{def1}\rm
 Let $X$ be a real Banach space, if for any sequence $\{u_n\}\subset X$ 
such that $f(u_n)$ is bounded and $f'(u_n)\to 0$ as $n\to\infty$, 
$\{u_n\}$ has a convergent subsequence, the functional $f$ satisfies the 
$(PS)$ condition.
\end{definition}

\begin{definition}[\cite{DR,ST}] \label{def2} \rm
 Let $Q$ be a submanifold of a Banach space $X$ with relative boundary 
$\partial Q, S$ be a closed subset of a Banach space $Y$ and $G$ be a subset 
of $C(\partial Q, Y\backslash S)$. $S$ and $\partial Q$ are G-linking if for any
map $h\in C(Q,Y)$ such that $h|_{\partial Q}\in G$ there holds $h(Q)\cap S\neq\emptyset$.
\end{definition}

\begin{theorem}[\cite{DR,ST}] \label{thmA}
 Let $X, Y$ be Banach spaces, $S$ be a closed subset of $Y$, $Q$ be a submanifold 
of $X$ with relative boundary $\partial Q$ and $G$ be a subset of
$C(\partial Q, Y\backslash S)$. Let $\Gamma=\{h\in C(Q,Y): h|_{\partial Q}\in G\}$, 
assume that $S$ and $\partial Q$ are G-linking and $f\in C^1(Y, R)$ satisfies
\begin{itemize}
\item[(a)] There is $\tilde{h}\in\Gamma$ such that
 $\sup_{x\in Q}f(\tilde{h}(x))<+\infty$;

\item[(b)] There is $\beta_0>\alpha_0$ such that
$$
\inf_{y\in S}f(y)\geq\beta_0\quad \text{and}\quad
\sup_{x\in\partial Q}f(h(x))\leq\alpha_0,\ \forall h\in \Gamma;
$$

\item[(c)] The $(PS)$ condition holds.
\end{itemize}
Then, the number
$$
c=\inf_{h\in\Gamma}\sup_{x\in Q}f(h(x))
$$
is a critical value of $f$ with $c\geq\beta_0$.
\end{theorem}

\begin{proof} 
The proof is divided into two steps.
\smallskip

\noindent\textbf{Step 1.} 
The $(PS)$ condition for the functional $J$ is satisfied. Let $(u_n,v_n)$ 
be a $(PS)$ sequence for the functional $J$; that is,
\begin{equation}\label{P10}
J(u_n,v_n)\text{ is bouned and }  J'(u_n,v_n)\to 0 \text{ as }n\to \infty.
\end{equation}
From (H1) and by a standard argument, it is sufficient to prove that 
$(u_n,v_n)$ is bounded in $W$. If this does not hold, assume that 
$\|(u_n,v_n)\|=\|u_n\|_p+\|v_n\|_q\to\infty$ as $n\to\infty$.
 Define $K_n:=\|u_n\|^p_p+\|v_n\|^q_q$, hence it follows that 
$K_n\to \infty$ as $n\to\infty$. 
Let $\bar{u}_n=u_n\setminus K_n^{1/p}, \bar{v}_n=v_n\setminus K_n^{1/q}$, 
then $(\bar{u}_n,\bar{v}_n)$ is bounded in $W$, i.e.,
$$
\|\bar{u}_n\|_p^p+\|\bar{v}_n\|_q^q=1 \quad \text{for all }n.
$$
Extracting subsequences if necessary, we can assume that there exists 
$(\bar{u},\bar{v})\in W$ such that
\begin{gather}\label{P11}
 (\bar{u}_n,\bar{v}_n)\rightharpoonup (\bar{u},\bar{v}) \quad\text{weakly in } W,\\
\label{P12}
(\bar{u}_n,\bar{v}_n)\to (\bar{u},\bar{v}) \quad\text{strongly in }
 L^p(\Omega)\times L^q(\Omega), \\
\label{P13}
(\bar{u}_n(x),\bar{v}_n(x))\to (\bar{u}(x),\bar{v}(x)) \quad 
 \text{for a.e. }x\in \Omega.
\end{gather} 
From \eqref{P10}, it follows that
\begin{equation}\label{P14}
\limsup_{n\to\infty}\frac{J(u_n,v_n)}{K_n}\leq0,
\end{equation}
From \eqref{P9}, (H1), the H\"older's inequality and \eqref{P3}, we have
\begin{equation} \label{P15}
\begin{aligned}
\big|\int_\Omega G(x,u,v)dx\big|
&\leq \int_\Omega\Big|\int_0^1(G_s(x,\tau u,\tau v)u+G_t(x,\tau
u,\tau v)v)d\tau\Big|dx \\
&\leq \int_\Omega h(x)(|u|+|v|)dx \\
&\leq  \|h\|_{L^\infty}(|\Omega|^\frac{p-1}{p}\|u\|_{L^p}
+|\Omega|^\frac{q-1}{q}\|v\|_{L^q}) \\
&\leq C_1(\|u\|_p+\|v\|_q)
\end{aligned}
\end{equation}
for all $(u,v)\in W$, where $C_1$ is a positive constant, hence it follows that
\begin{equation}\label{P16}
\frac{1}{K_n}\int_\Omega G(x,u_n,v_n)dx\to 0\quad \text{as }n\to\infty.
\end{equation}

From $h_1\in L^{p/(p-1)}(\Omega), h_2\in
L^{q/(q-1)}(\Omega)$ and the H\"older's inequality, we obtain
\begin{equation}\label{P17}
\frac{1}{K_n}\int_\Omega (h_1u_n+h_2v_n)dx\to 0\quad 
\text{as }n\to\infty.
\end{equation}

From \eqref{P12} and \eqref{P13}, it follows that 
$|\bar{u}_n|^{\alpha}\bar{u}_n\to|\bar{u}|^\alpha\bar{u}$ strongly
in $L^{\frac{p}{\alpha+1}}(\Omega)$ and
$|\bar{v}_n|^{\beta}\bar{v}_n\to|\bar{v}|^\alpha\bar{v}$ strongly in
$L^{\frac{q}{\beta+1}}(\Omega)$, hence from H\"older's inequality, we
obtain
\begin{equation}\label{P18}
\begin{aligned}
&\Big|\int_{\Omega}(|\bar{u}_n|^{\alpha}
|\bar{v}_n|^{\beta}\bar{u}_n\bar{v}_n-
|\bar{u}|^{\alpha}|\bar{v}|^{\beta}\bar{u}\bar{v})dx\Big|  \\
&\leq \int_{\Omega}||\bar{u}_n|^{\alpha}
|\bar{v}_n|^{\beta}\bar{u}_n\bar{v}_n-
|\bar{u}_n|^{\alpha}|\bar{v}|^{\beta}\bar{u}_n\bar{v}|dx
+\int_{\Omega}||\bar{u}_n|^{\alpha}
|\bar{v}|^{\beta}\bar{u}_n\bar{v}-
|\bar{u}|^{\alpha}|\bar{v}|^{\beta}\bar{u}\bar{v}|dx  \\
&\leq \int_{\Omega}|\bar{u}_n|^{\alpha+1}\cdot|
|\bar{v}_n|^{\beta}\bar{v}_n-|\bar{v}|^{\beta}\bar{v}|dx
+\int_{\Omega}||\bar{u}_n|^{\alpha}\bar{u}_n-
|\bar{u}|^{\alpha}\bar{u}|\cdot|\bar{v}|^{\beta+1}dx \\
&\leq \|\bar{u}_n\|_{L^p}^{\alpha+1}\cdot\|
|\bar{v}_n|^{\beta}\bar{v}_n-|\bar{v}|^{\beta}\bar{v}\|_{L^{\frac{q}{\beta+1}}}
+\|\bar{v}_n\|_{L^q}^{\beta+1}\cdot\|
|\bar{u}_n|^{\alpha}\bar{u}_n-|\bar{u}|^{\alpha}\bar{u}\|_{L^{\frac{p}{\alpha+1}}}
 \\ 
&\to 0\quad \text{as }n\to\infty.
\end{aligned}
\end{equation}

From the definition of $J$, \eqref{P12}, \eqref{P14}, \eqref{P16}, \eqref{P17}
and \eqref{P18}, we have
\begin{align*}
&\limsup_{n\to\infty}\Big(\frac{1}{p}\int_\Omega|\nabla
\bar{u}_n|^pdx+\frac{1}{q}\int_\Omega|\nabla \bar{v}_n|^qdx\Big)  \\
&\leq \lambda_1\Big(\frac{1}{p}\int_\Omega|\bar{u}|^pdx
+\frac{1}{q}\int_\Omega|\bar{v}|^qdx+
\frac{1}{(\alpha+1)(\beta+1)}\int_{\Omega}|\bar{u}|^{\alpha}|\bar{v}|^{\beta}
\bar{u}\bar{v}dx\Big).
\end{align*}
From \eqref{P11}, it follows that
$$
\int_\Omega|\nabla \bar{u}|^pdx\leq
\liminf_{n\to\infty}\int_\Omega|\nabla\bar{u}_n|^pdx,\quad
\int_\Omega|\nabla \bar{v}|^qdx\leq\liminf_{n\to\infty}
\int_\Omega|\nabla \bar{v}_n|^qdx,
$$
hence, combining this with the definition of $\lambda_1$, we obtain
\begin{align*}
&\lambda_1\Big(\frac{1}{p}\int_\Omega|\bar{u}|^pdx
+\frac{1}{q}\int_\Omega|\bar{v}|^qdx+\frac{1}{(\alpha+1)(\beta+1)}\int
_{\Omega}|\bar{u}|^{\alpha}|\bar{v}|^{\beta}
\bar{u}\bar{v}dx\Big) \\
&\leq \frac{1}{p}\int_\Omega|\nabla \bar{u}|^pdx
+\frac{1}{q}\int_\Omega|\nabla \bar{v}|^qdx \\
&\leq \liminf_{n\to\infty}\Big(\frac{1}{p}\int_\Omega|\nabla\bar{u}_n|^pdx
+\frac{1}{q}\int_\Omega|\nabla \bar{v}_n|^qdx\Big)  \\
 &\leq \limsup_{n\to\infty}\Big(\frac{1}{p}\int_\Omega|\nabla\bar{u}_n|^pdx
+\frac{1}{q}\int_\Omega|\nabla \bar{v}_n|^qdx\Big)  \\
 &\leq \lambda_1\Big(\frac{1}{p}\int_\Omega|\bar{u}|^pdx
+\frac{1}{q}\int_\Omega
|\bar{v}|^qdx+\frac{1}{(\alpha+1)(\beta+1)}\int
_{\Omega}|\bar{u}|^{\alpha}|\bar{v}|^{\beta}
\bar{u}\bar{v}dx\Big),
\end{align*}
hence it follows that
\begin{align*}
&\frac{1}{p}\int_\Omega|\nabla \bar{u}|^pdx
+\frac{1}{q}\int_\Omega|\nabla \bar{v}|^qdx \\
 &= \lambda_1\Big(\frac{1}{p}\int_\Omega|\bar{u}|^pdx
+\frac{1}{q}\int_\Omega|\bar{v}|^qdx+\frac{1}{(\alpha+1)(\beta+1)}\int
_{\Omega}|\bar{u}|^{\alpha}|\bar{v}|^{\beta}
\bar{u}\bar{v}dx\Big),
\end{align*}
and by the uniform convexity of $W$, we have that $(\bar{u}_n,\bar{v}_n)$ 
converges strongly to $(\bar{u},\bar{v})$ in $W$, and from the definition of
$(\mu_1,\nu_1)$, it follows that $(\bar{u},\bar{v})=\pm(\mu_1,\nu_1)$.

In the following, let $(\bar{u},\bar{v})=(\mu_1,\nu_1)$, and the other case where
$(\bar{u},\bar{v})=-(\mu_1,\nu_1)$ may be considered similarly. 
Hence from the definition of $J$, we have
\begin{equation} \label{P19}
\begin{aligned}
&\frac{pJ_1(u_n,v_n)}{(p-1)K_n^{1/p}}+
\frac{qJ_2(u_n,v_n)}{(q-1)K_n^{1/q}}-\langle J'(u_n,v_n),
(\frac{\bar{u}_n}{p-1},\frac{\bar{v}_n}{q-1})\rangle  \\
&= \frac{1}{p-1}\Big(\int_\Omega G_s(x,u_n,v_n)\bar{u}_n dx-
\frac{p}{K_n^{1/p}}\int_\Omega\int_0^1G_s(x,ru_n,rv_n)u_n\,dr\,dx\Big)  \\
&\quad +\frac{1}{q-1}\Big(\int_\Omega G_t(x,u_n,v_n)\bar{v}_ndx-
\frac{q}{K_n^{1/q}}\int_\Omega\int_0^1G_t(x,ru_n,rv_n)v_n\,dr\,dx\Big)  \\
&\quad +\int_\Omega h_1\bar{u}_ndx+\int_\Omega h_2\bar{v}_ndx.
\end{aligned}
\end{equation}
From $h_1\in L^{p/(p-1)}(\Omega), h_2\in L^{q/(q-1)}(\Omega)$, we have
\begin{equation}\label{P20}
\int_\Omega h_1\bar{u}_ndx\to\int_\Omega h_1\mu_1dx\ \text{ and }\
\int_\Omega h_2\bar{v}_ndx\to\int_\Omega  h_2\nu_1 dx\ \ \text{ as
}\ n\to\infty.
\end{equation}
From (H2) and (H3), it is easy to know that
\begin{equation}\label{P21}
\begin{gathered}
\int_\Omega G_s(x,u_n,v_n)\bar{u}_ndx\to\int_\Omega g_1^{++}\mu_1dx,\\
\int_\Omega G_t(x,u_n,v_n)\bar{v}_ndx \to\int_\Omega
g_2^{++}\nu_1dx
\end{gathered}
\end{equation}
as $n\to\infty$. Finally, from (H2) and Lebesgue dominated convergence
theorem, we have
\begin{equation} \label{P22}
\begin{aligned}
\frac{1}{K_n^{1/p}}\int_\Omega\int_0^1G_s(x,ru_n,rv_n)u_n\,dr\,dx
&= \int_\Omega\int_0^1G_s(x,ru_n,rv_n)\frac{u_n}{K_n^{1/p}}\,dr\,dx \\
&\to \int_\Omega g_1^{++}\mu_1dx \quad \text{as } n\to\infty.
\end{aligned}
\end{equation}
Similarly, we obtain
\begin{equation}\label{P23}
\frac{1}{K_n^{1/q}}\int_\Omega\int_0^1G_t(x,ru_n,rv_n)v_n\,dr\,dx
\to\int_\Omega g_2^{++}\nu_1dx \quad \text{as } n\to\infty.
\end{equation}
Therefore, letting $n\to \infty$ in \eqref{P19} and from \eqref{P10},
\eqref{P20}, \eqref{P21}, \eqref{P22} and \eqref{P23}, we obtain
$$
\int_\Omega h_1\mu_1dx+\int_\Omega h_2\nu_1dx=\int_\Omega
g_1^{++}\mu_1dx+\int_\Omega g_2^{++}\nu_1dx,
$$
which contradicts with \eqref{P5}. Hence, $(u_n,v_n)$ is bounded in $W$.
\smallskip

\noindent\textbf{Step 2.}
The functional $J$ satisfies the geometries of Theorem \ref{thmA}.
For any 
\[
(u,v)\in E_1=\{(t^{1/p}\mu_1,t^{1/q}\nu_1):t\geq0\}
\cup\{(-t^{1/p}\mu_1,-t^{1/q}\nu_1):t\geq0\},
\]
 we have
\begin{align*}
&\frac{1}{p}\int_\Omega|\nabla u|^pdx+\frac{1}{q}\int_\Omega|\nabla v|^q\,dx\\
&= \lambda_1\Big(\frac{1}{p}\int_\Omega |u|^pdx+\frac{1}{q}\int_\Omega |v|^qdx
+\frac{1}{(\alpha+1)(\beta+1)}\int_{\Omega}|u|^{\alpha}|v|^{\beta}uv\,dx\Big).
\end{align*}
From the above equality and the definition of $J$, for any
$(t^{1/p}\mu_1,t^{1/q}\nu_1)\in E_1$, we obtain
\begin{equation}\label{P24}
\begin{aligned}
&J(t^{1/p}\mu_1,t^{1/q}\nu_1)\\
&= t^{1/p}\int_\Omega h_1\mu_1dx+t^{1/q}\int_\Omega h_2\nu_1dx
 -\int_\Omega G(x,t^{1/p}\mu_1,t^{1/q}\nu_1)dx \\
&= t^{1/p}\Big(\int_\Omega h_1\mu_1dx
 -\int_\Omega\int_0^1G_s(x,r t^{1/p}\mu_1,r t^{1/q}\nu_1)\mu_1\,dr\,dx\Big) \\
&\quad +t^{1/q}\Big(\int_\Omega h_2\nu_1dx
 -\int_\Omega\int_0^1G_t(x,r t^{1/p}\mu_1,r t^{1/q}\nu_1)\nu_1\,dr\,dx\Big).
\end{aligned}
\end{equation}

From (H1), (H2) and Lebesgue dominated convergence theorem, it follows that
\begin{gather}\label{P25}
\lim_{t\to+\infty}\int_\Omega\int_0^1G_s(x,r t^{1/p}\mu_1,r t^{1/q}\nu_1 )
\mu_1\,dr\,dx
=\int_\Omega g_1^{++}\mu_1dx, \\
\label{P26}
\lim_{t\to+\infty}\int_\Omega\int_0^1G_t(x,rt^{1/q}\mu_1,rt^{1/q}\nu_1)\nu_1\,dr\,dx
=\int_\Omega g_2^{++}\nu_1dx.
\end{gather}
Hence, from \eqref{P7}, \eqref{P24}, \eqref{P25} and \eqref{P26}, we obtain
\begin{align*}
J(t^{1/p}\mu_1,t^{1/q}\nu_1)\to-\infty\quad  \text{as } t\to+\infty.
\end{align*}
Similarly, the following result can be obtained
with $g_1^{++}$ and $g_2^{++}$ exchanged with $g_1^{--}$ and $g_2^{--}$
respectively,
\[
J(-t^{1/p}\mu_1,-t^{1/q}\nu_1)\to-\infty\quad \text{as } t\to+\infty.
\]
Finally, it follows that
\begin{equation}\label{P27}
\lim_{|t|\to\infty}J(\pm t^{1/p}\mu_1,\pm t^{1/q}\nu_1)=-\infty.
\end{equation}

On the other hand, letting $\Lambda_2:=\{(u,v)\in W: \phi(u,v)\geq
\lambda_2\varphi(u,v)\}$, from \eqref{P3}, \eqref{P15}
and the H\"older's inequality, for any $(u,v)\in \Lambda_2$,
we obtain
\begin{align*}
J(u,v)&\geq \frac{\lambda_2-\lambda_1}{p\lambda_2}
\|u\|_p^p+\frac{\lambda_2-\lambda_1}{q\lambda_2}\|v\|_q^q
-C_1(\|u\|_p+\|v\|_q) \\
&\quad -(\|h_1\|_{L^{\frac{p}{p-1}}}\|u\|_{L^p}+
\|h_2\|_{L^{\frac{q}{q-1}}}\|v\|_{L^q}) \\
&\geq \frac{\lambda_2-\lambda_1}{p\lambda_2}
(\|u\|_p^p+\|v\|_q^q)
-C_2(\|u\|_p+\|v\|_q),
\end{align*}
where $C_2=C_1+C\max\{\|h_1\|_{L^{\frac{p}{p-1}}}, \|h_2\|_{L^{\frac{q}{q-1}}}\}$. 
Combining the above expression with \eqref{P27}, we obtain that there exists a
positive constant $T$ such that
\begin{equation}\label{P28}
\alpha_0:=\sup_{t\geq T}J(\pm t^{1/p}\mu_1,\pm
t^{1/q}\nu_1)<\beta_0:=\inf_{(u,v)\in \Lambda_2}J(u,v).
\end{equation}

Let $M=\{(\pm t^{1/p}\mu_1,\pm t^{1/q}\nu_1):t\geq T\}$ and 
\[
G=\{h\in C(S^0,W):h\text{ is odd and } h(S^0)\subset M\},
\]
 where $S^0$ 
is the boundary of the closed unit ball $B^1$ in $\mathbb{R}^1$, 
i.e., $S^0=\partial B^1$. For any $h\in G$, by \eqref{P28}, we have
$h(S^0)\cap\Lambda_2=\emptyset$, which implies that $G$ is a subset of
$C(S^0,W\setminus\Lambda_2)$. Let
$$
\Gamma=\{h\in C(B^1,W): h|_{S^0}\in G\},
$$
we can claim: $\Gamma$ is nonempty and $\Lambda_2$ and $S^0$ 
are G-linking, that is $h(B^1)\cap\Lambda_2\neq\emptyset$ for any 
$h\in \Gamma$. The similar proof of the conclusion may be found 
in \cite{OT,DR,ST}, but for the reader¡¯s convenience and completeness, 
we write it.

  In fact, define $\bar{h}:B^1\to W$ by
\begin{gather*}
\bar{h}(t)=((tT)^{1/p}\mu_1,(tT)^{1/q}\nu_1)\quad \text{for all }t\in[0,1], \\
\bar{h}(-t)=(-(tT)^{1/p}\mu_1,-(tT)^{1/q}\nu_1)\quad  \text{for all }t\in[0,1].
\end{gather*}
Hence, $\bar{h}\in\Gamma$ and $\Gamma$ is nonempty. Now let $h\in\Gamma$, 
if there is $(u,v)\in h(B^1)$ such that $\varphi(u,v)=0$, we get 
$h(B^1)\cap\Lambda_2\neq\emptyset$. If not, we consider the map
$\hat{h}:S^1\to \Sigma$ defined by
$$
\hat{h}(x_1, x_2)= \begin{cases}
 \pi\circ h(x_1), &\text{if } x_2\geq0, \\
-\pi\circ h(-x_1), &\text{if } x_2\leq0,
\end{cases}
$$
where $\pi(u,v)=(u\setminus(\varphi(u,v))^{1/p}, v\setminus(\varphi(u,v))^{1/q})$. 
It is easy to know that $\hat{h}(S^1)\subset\Sigma_2$. Therefore, 
$\phi(u_0,v_0)\geq\lambda_2$
for some $(u_0,v_0)\in \hat{h}(S^1)$, namely, $(u_0,v_0)\in\Lambda_2$. 
From $\pi\circ h(x)\in\Lambda_2$, we have implies $h(x)\in\Lambda_2$, 
which implies that $h(B^1)\cap\Lambda_2\neq\emptyset$. Hence $\Lambda_2$ and
$S^0$ are G-linking.

Now, from the compactness of $B^1$, (a) of Theorem \ref{thmA} holds, (b) 
of Theorem \ref{thmA}
is satisfied from \eqref{P28}, (c) of Theorem \ref{thmA} comes from (i). Accordingly,
 Theorem \ref{thm1} holds from the G-linking Theorem with the critical value
$$
c=\inf_{h\in\Gamma}\sup_{x\in B^1}J(h(x)).\ \ \ \Box
$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
 (i) The functional $J$ satisfies the $(PS)$ condition. From $\eqref{P6}$, 
the claim can be proved with similar to step 1 of Theorem \ref{thm1}.

(ii) Now we will prove that the functional $J$ is coercive, that is,
$$
J(u,v)\to+\infty\quad \text{as }\ \|(u,v)\|\to\infty.
$$
If the claim does not hold, there is a constant $c$ and a sequence $(u_n,v_n)$ 
such that $J(u_n,v_n)\leq c$ and $\|(u_n,v_n)\|\to\infty$. 
From the proof of the (PS) condition of
Theorem \ref{thm1}, $(\bar{u}_n,\bar{v}_n)$ converges strongly to $\pm(\mu_1,\nu_1)$, 
where $\bar{u}_n=u_n\setminus K_n^{1/p}, \bar{v}_n=v_n\setminus K_n^{1/q}$. 
Assume that $(\bar{u}_n,\bar{v}_n)$ converges strongly to $(\mu_1,\nu_1)$
(the case $(\bar{u}_n,\bar{v}_n)$ converges strongly to $(-\mu_1,-\nu_1)$ 
may be treated similarly) and $p\geq q>1$ (the case $q\geq p>1$ may also be 
treated similarly). From the definitions of $J,J_1,J_2$ and 
$J(u_n,v_n)\leq c$ for all $n$, \eqref{P22} and \eqref{P23}, we have
\begin{align*}
0&\geq \limsup_{n\to\infty}\frac{J(u_n,v_n)}{K_n^{1/p}}
=\limsup_{n\to\infty}\Big(\frac{J_1(u_n,v_n)}{K_n^{1/p}}
+\frac{J_2(u_n,v_n)}{K_n^{1/p}}\Big) \\
&\geq \limsup_{n\to\infty}\Big(\frac{J_1(u_n,v_n)}{K_n^{1/p}}+
\frac{J_2(u_n,v_n)}{K_n^{1/q}}\Big) \\
&\geq \lim_{n\to\infty}\Big(\int_\Omega
(h_1\bar{u}_n-\frac{1}{K_n^{1/p}}\int_0^1G_s(x,ru_n,rv_n)u_ndr)dx  \\
&\quad +\int_\Omega(h_2\bar{v}_n-
\frac{1}{K_n^{1/q}}\int_0^1G_t(x,ru_n,rv_n)v_ndr)dx\Big) \\
&= \int_\Omega(h_1\mu_1-g_1^{++}\mu_1)dx
+\int_\Omega(h_2\nu_1-g_2^{++}\nu_1)dx,
\end{align*}
which is a contradiction to \eqref{P6}.
 By Ekeland's Variational Principle, the proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by National
Natural Science Foundation of China (No. 11471267),  by the
Fundamental Research Funds for the Central Universities (No.
XDJK2011C039), and by the Postdoctoral Research Foundation of Chongqing
(No. Xm201319).

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\end{document}