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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 47, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/47\hfil Multiplicity of positive solutions]
{Multiplicity of positive solutions for a Navier boundary-value 
problem involving the $p$-biharmonic with critical exponent}

\author[Y. Shen, J. Zhang\hfil EJDE-2011/47\hfilneg]
{Ying Shen, Jihui Zhang}  % in alphabetical order

\address{Ying Shen \newline
Jiangsu Key Laboratory for NSLSCS,
School of Mathematical Sciences,
Nanjing Normal University, 210046, Jiangsu, China}
\email{shenying99@126.com}

\address{Jihui Zhang \newline
Jiangsu Key Laboratory for NSLSCS,
School of Mathematical Sciences,
Nanjing Normal University, 210046, Jiangsu, China}
\email{jihuiz@jlonline.com}

\thanks{Submitted December 18, 2010. Published April 6, 2011.}
\subjclass[2000]{35J40, 35J67}
\keywords{$p$-biharmonic system;
Navier condition; Nehari manifold; \hfill\break\indent
critical exponent}

\begin{abstract}
 By using the Nehari manifold and variational methods, we prove
 that a $p$-biharmonic system has at least two positive solutions
 when the pair the parameters satisfy certain inequality.
\end{abstract}

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

\section{Introduction}

In this article, we consider the multiplicity results of positive
solutions of the  semilinear  p-biharmonic system
\begin{equation}\label{e1.1}
\begin{gathered}
\Delta(|\Delta u|^{p-2}\Delta u)=\frac{1}{p^{**}}
 \frac{\partial F(x,u,v)}{\partial u}+\lambda |u|^{q-2}u
\quad\text{in }  \Omega,  \\
\Delta(|\Delta v|^{p-2}\Delta v)
=\frac{1}{p^{**}}\frac{\partial F(x,u,v)}{\partial v}
+\mu |v|^{q-2}v \quad\text{in }  \Omega,   \\
u>0,\quad v>0  \quad\text{in }  \Omega,   \\
u=v=\Delta u=\Delta v=0 \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $x_0\in \Omega$ is a bounded domain in $\mathbb{R}^N$ with
smooth boundary $\partial \Omega$,
$F\in C^{1}(\overline{\Omega}\times (\mathbb{R}^+)^2, \mathbb{R}^+)$
is positively homogeneous of degree
$p^{**}=\frac{pN}{N-2p}$ which is the  Sobolev
critical exponent; that is,
$F(x,tu,tv)=t^{p^{**}}F(x,u,v)$ $(t>0)$ holds for all
$(x,u,v)\in \overline{\Omega}\times (\mathbb{R}^+)^2$,
$(\frac{\partial F(x,u,v)}{\partial u},
\frac{\partial F(x,u,v)}{\partial v})=\nabla F$. We assume that
$1<q<p<\frac{N}{2 }$, $\lambda>0$, $\mu>0$.

 In recent years, there have been many article concerned with the
existence and multiplicity of positive solutions for $p$-biharmonic
elliptic problems. Results relating to these problems can be found
in \cite{c1,d1,e2,l1,t1,t2,w1,w2} and the references therein.

 Brown and Wu \cite{b2} considered the  semilinear
elliptic system
\begin{equation}\label{e1.2}
\begin{gathered}
 -\Delta u+u=\frac{\alpha}{\alpha+\beta}f(x)|u|^{\alpha-2}u|v|^{\beta}
 \quad\text{in }  \Omega,  \\
-\Delta v+v=\frac{\beta}{\alpha+\beta}f(x)|u|^{\alpha}|v|^{\beta-2}v
 \quad\text{in }  \Omega,   \\
\frac{\partial u}{\partial n}=\lambda g(x)|u|^{q-2}u,\quad
\frac{\partial v}{\partial n}=\mu h(x)|v|^{q-2}v\quad
\text{on }\partial \Omega.
\end{gathered}
\end{equation}
where $\alpha>1$, $\beta>1$ satisfying $2<\alpha+\beta<2^{*}$ and the
weight functions $f,g,h$ are satisfying the following conditions:
\begin{itemize}
\item[(A)] $f\in C(\overline{\Omega})$ with $\|f\|_{\infty}=1$ and
$f^{+}=\max\{f,0\}\not \equiv 0$;

\item[(B)] $g,h\in C(\partial \Omega)$ with
$\|g\|_{\infty}=\|h\|_{\infty}=1,\ g^{\pm}=\max\{\pm g,0\}\not
\equiv 0$ and $h^{\pm}=\max\{\pm h,0\}\not \equiv 0$.
\end{itemize}
They showed that \eqref{e1.2} has at least two negative solutions if
the pair of the parameters $(\lambda, \mu)$ belongs to a certain
subset of $\mathbb{R}^2$.

 Recently, Hsu \cite{h1}  considered the case
$F(x,u,v)=2|u|^{\alpha}|v|^{\beta},\alpha>1,\beta>1$ satisfying
$\alpha+\beta=p^{*}$; i.e., the elliptic system:
\begin{equation}\label{e1.3}
\begin{gathered}
 -\Delta_{p} u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}
+\lambda |u|^{q-2}u  \quad\text{in }  \Omega,  \\
-\Delta_{p} v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v
+\mu |v|^{q-2}v  \quad\text{in }  \Omega,   \\
u=v=0\quad \text{on }\partial \Omega.
\end{gathered}
\end{equation}
By variational methods, he proved that  \eqref{e1.2} has at least
two positive solutions if the pair of the parameters $(\lambda,\mu)$
belongs to a certain subset of $\mathbb{R}^2$.

 In this article, we give a  simple variational method
which is similar to the ``fibering method'' of Pohozaev's (
see \cite{d2,b4}) to prove the existence of at least two positive solutions
of problem \eqref{e1.1}.
Throughout this paper, we let $S$ be the best
Sobolev embedding constant defined by
\[
S=\inf_{u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)\backslash
\{0\}}\frac{\int_{\Omega}|\Delta
u|^pdx}{(\int_{\Omega}|u|^{p^{**}}dx)^{\frac{p}{p^{**}}}},
\]
 and let
\begin{gather*}
C(p,q,N,K,S,|\Omega|)=(\frac{p-q}{K(p^{**}-q)})^{\frac{p}{p^{**}-q}}
(\frac{p^{**}-q}{p^{**}-p}|\Omega|^{\frac{p^{**}-q}{p^{**}}})
^{-\frac{p}{p-q}}S^{\frac{N}{2p}+\frac{q}{p-q}},\\
C_0=(\frac{q}{p})^{\frac{p}{p-q}}C(p,q,N,K,S,|\Omega|).
\end{gather*}

For our results, we need the following assumptions:
\begin{itemize}
\item[(F1)]
$F:\overline{\Omega}\times \mathbb{R}^+\times \mathbb{R}^+
\to\mathbb{R}^+$ is a $C^{1}$ function and
$F(x,tu,tv)=t^{p^{**}}F(x,u,v)$ for all $t>0$ and
$x\in \overline{\Omega}$, $(u,v)\in (\mathbb{R}^+)^2$;

\item[(F2)]
$F(x,u,0)=F(x,0,v)=\frac{\partial F}{\partial
u}(x,u,0)=\frac{\partial F}{\partial v}(x,0,v)=0$,
where $u,v\in \mathbb{R}^+$;

\item[(F3)]
$\frac{\partial F(x,u,v)}{\partial u}, \frac{\partial
F(x,u,v)}{\partial v}$ are strictly increasing functions about $u$
and $v$ for all $u>0$, $v>0$.
\end{itemize}

From assumption (F1), we have the
so-called Euler identity
\begin{equation}\label{e1.4}
(u,v)\cdot\nabla F(x,u,v)=p^{**} F(x,u,v)
\end{equation}
and, for a positive constant $K$,
\begin{equation}\label{e1.5}
F(x,u,v)\leq K(|u|^p+|v|^p)^{\frac{p^{**}}{p}}.
\end{equation}

\begin{theorem} \label{thm1.1}
If $\lambda,\mu$ satisfy
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C(p,q,N,K,S,|\Omega|)$,
and {\rm(F1)--(F3)} hold, then \eqref{e1.1} has at least one
positive solution.
\end{theorem}

\begin{theorem} \label{thm1.2}
If  $\lambda,\mu$ satisfy
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0^{*}$,
 {\rm (F1)--(F3)} hold, where $C_0^{*}=\min\{C^{*},C_0\}$, and
$C^{*}=\min\{\delta_{1},\rho_0^{\frac{N-2p}{p-1}},\delta_2\}$,
then \eqref{e1.1} has at least two
positive solutions.
\end{theorem}

\begin{remark} \label{rmk1.3} \rm
There are functions  satisfying the
conditions of  Theorems \ref{thm1.1} and \ref{thm1.2}. For example,
\[
F(x,u,v)=\begin{cases}
f_{1}^2(x)|u|^{3/2}|v|^{5/2}+f_2^2(x)\frac{u^3v^3}{u^2+v^2}
 &\text{if } (u,v)\neq(0,0),  \\
0&\text{if } (u,v)=(0,0),
\end{cases}
\]
where $f_{1}, f_2\in C(\overline{\Omega})\cap L^{\infty}(\Omega)$
with $\max\{\pm f_{1},\pm f_2,0\}\not\equiv0 $.
Obviously, $F$ satisfy (F1), (F2) and (F3).
\end{remark}

This article is organized as follows:
In Section 2, we give some notation and preliminaries.
In Section 3, we prove Theorems \ref{thm1.1} and \ref{thm1.2}.

\section{Notation and preliminaries}

Problem \eqref{e1.1} is posed in the framework of the Sobolev space
$E=(W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))
\times (W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega))$
with the standard norm
\[
\|(u,v)\|^p=\int_{\Omega}|\Delta u|^pdx+\int_{\Omega}|\Delta
v|^pdx=\|\Delta u\|_{L^p(\Omega)}^p+\|\Delta
v\|_{L^p(\Omega)}^p.
\]
In addition, we define
$\|u\|_{L^p(\Omega)}=(\int_{\Omega}|u|^pdx)^{\frac{1}{p}}$
as the norm of the Sobolev space $L^p(\Omega)$.

A pair of functions $(u^{+},v^{+})\in E$, with
$(u^{+}:=\max\{u,0\}$ and $v^{+} :=\max\{v,0\})$,
is said to be a weak solution of \eqref{e1.1} if
\begin{align*}
&\int_{\Omega}(|\Delta u^{+}|^{p-2}\Delta
u^{+}\Delta\varphi_{1}+|\Delta v^{+}|^{p-2}\Delta
v^{+}\Delta\varphi_2)dx-
\frac{1}{p^{**}}\int_{\Omega}\frac{\partial
F(x,u^{+},v^{+})}{\partial u}\varphi_{1}dx\\
&- \frac{1}{p^{**}}\int_{\Omega}\frac{\partial
F(x,u^{+},v^{+})}{\partial v}\varphi_2dx -
\lambda\int_{\Omega}|u^{+}|^{q-2}u\varphi_{1}dx-\mu\int_{\Omega}|v^{+}|^{q-2}v\varphi_2dx
= 0
\end{align*}
for all $(\varphi_{1},\varphi_2)\in E$. Thus, by \eqref{e1.4} the
corresponding energy functional of problem \eqref{e1.1} is
defined by
\[
J_{\lambda,\mu}(u^{+},v^{+})=\frac{1}{p}\|(u^{+},v^{+})\|^p
-\frac{1}{p^{**}}\int_{\Omega}F(x,u^{+},v^{+})dx
-\frac{1}{q}K_{\lambda,\mu}(u^{+},v^{+})
\]
for $(u^{+},v^{+})\in E$, where
$K_{\lambda,\mu}(u^{+},v^{+})=\lambda\int_{\Omega}|u^{+}|^{q}dx
+\mu\int_{\Omega}|v^{+}|^{q}dx$.

 To verify $J_{\lambda,\mu}\in C^{1}(E,R)$, we
need the  following lemmas.

\begin{lemma} \label{lem2.1}
Suppose that (F3) holds. Assume that
$F\in C^{1}(\overline{\Omega}\times (\mathbb{R}^+)^2, \mathbb{R}^+)$
is positively homogeneous of degree $p^{**}$, then
$\frac{\partial F}{\partial u},\frac{\partial F}{\partial v}\in
C(\overline{\Omega}\times (\mathbb{R}^+)^2, \mathbb{R}^+)$ are
positively homogeneous of degree $p^{**}-1$.
\end{lemma}

The proof of the above lemma is almost the same as that in
Chu and Tang \cite{c2}, and it is omitted.

  From Lemma \ref{lem2.1}, we obtain the existence of a
positive constant $M$ such that for all $x\in \overline{\Omega}$,
\begin{gather} \label{e2.1}
\big|\frac{\partial F}{\partial u}(x,u,v)\big|\leq
M(|u|^{p^{**}-1}+|v|^{p^{**}-1}), \\
\label{e2.2}
\big|\frac{\partial F}{\partial v}(x,u,v)\big|\leq
M(|u|^{p^{**}-1}+|v|^{p^{**}-1}),
u,v\in \mathbb{R}^+.
\end{gather}
As in Willem \cite[Theorem A.2]{w2}, we
consider the continuity of the superposition operator
\[
A:L^p(\Omega)\times L^p(\Omega)\to L^{q}(\Omega):
(u,v) \mapsto f(x,u,v).
\]

\begin{lemma} \label{lem2.2}
Assume that $|\Omega|<\infty$, $1\leq p$, $r<\infty$,
$f\in C(\overline{\Omega}\times \mathbb{R}^2,\mathbb{R})$ and
\[
 |f(x,u,v)|\leq c(1+|u|^{\frac{p}{r}}+|v|^{\frac{p}{r}}).
\]
 Then, for every $(u,v)\in
L^p(\Omega)\times L^p(\Omega)$, $f(\cdot,u,v)\in L^{r}(\Omega)$
and the operator
$A:L^p(\Omega)\times L^p(\Omega)\to L^{r}(\Omega)$:
$(u,v) \mapsto f(x,u,v)$ is continuous.
\end{lemma}

 Now we consider the functional
$\psi(u,v)=\int_{\Omega}F(x,u,v)dx$.

\begin{lemma} \label{lem2.3}
Assume that $|\Omega|<\infty$,
$\frac{\partial F}{\partial u}$,
$\frac{\partial F}{\partial v} \in C(\overline{\Omega}
\times (\mathbb{R}^+)^2)$ satisfying \eqref{e2.1}, \eqref{e2.2},
then the functional $\psi$ is of class $C^{1}(E,\mathbb{R}^+)$ and
\[
\langle\psi'(u,v),(a,b)\rangle=\int_{\Omega}(\frac{\partial
F(x,u,v)}{\partial u}a+\frac{\partial F(x,u,v)}{\partial v}b)dx,
\]
where $(u,v),(a,b)\in E$.
\end{lemma}

\begin{proof}
First, we proof the existence of the Gateaux
derivative. Given $x\in \Omega$ and $0<|t|<1$, by the mean value
theorem and \eqref{e2.1}, \eqref{e2.2}, there exists
$\lambda_{1}\in [0,1]$ such that
\begin{align*}
&\frac{|F(x,u+ta,v+tb)-F(x,u,v)|}{|t|} \\
&= |\frac{\partial F(x,u+t\lambda_{1}a,v
+t\lambda_{1}b)}{\partial u}a|
+ |\frac{\partial F(x,u+t\lambda_{1}a,v+t\lambda_{1}b)}{\partial v}b|\\
& \leq M(|u+a|^{p^{**}-1}+|v+b|^{p^{**}-1})|a|+M(|u+a|^{p^{**}-1}
 +|v+b|^{p^{**}-1})|b| \\
&\leq 2^{p^{**}-2}M(|u|^{p^{**}-1}+|v|^{p^{**}-1}
 +|a|^{p^{**}-1}+|b|^{p^{**}-1})(|a|+|b|).
\end{align*}
The  H\"older inequality and the Sobolev imbedding theorem imply that
\[
(|u|^{p^{**}-1}+|v|^{p^{**}-1}+|a|^{p^{**}-1}+|b|^{p^{**}-1})(|a|+|b|)
\in L^{1}(\Omega).
\]
It follows from the Lebesgue theorem that
\[
\langle\psi'(u,v),(a,b)\rangle=\int_{\Omega}(\frac{\partial
F(x,u,v)}{\partial u}a+\frac{\partial F(x,u,v)}{\partial v}b)\,dx.
\]
Next, we proof the continuity of the Gateaux derivative. Assume that
$(u_{n},v_{n})\to(u,v)$ in $E$. By Sobolev imbedding
theorem, $(u_{n},v_{n})\to(u,v)$ in
$L^{p^{**}}(\Omega)\times L^{p^{**}}(\Omega)$.
By Lemma \ref{lem2.2}, we
obtain that $\nabla F(x,u_{n},v_{n})\to \nabla F(x,u,v)$ in
$L^{\beta}(\Omega)$ where $\beta:=\frac{p^{**}}{p^{**}-1}$.
By the H\"older inequality and Sobolev imbedding theorem,
\begin{align*}
|\langle\psi'(u_{n},v_{n})-\psi'(u,v),(a,b)\rangle|
&\leq  \|\frac{\partial F(x,u_{n},v_{n})}{\partial u}-\frac{\partial
F(x,u,v)}{\partial u}\|_{L^{\beta}(\Omega)} \|
a\|_{L^{p^{**}}(\Omega)}\\
&\quad +\|\frac{\partial F(x,u_{n},v_{n})}{\partial v}-\frac{\partial
F(x,u,v)}{\partial v}\|_{L^{\beta}(\Omega)} \|
b\|_{L^{p^{**}}(\Omega)}\\
&\leq  S^{-\frac{1}{p}}(\|\frac{\partial F(x,u_{n},v_{n})}{\partial
u}-\frac{\partial F(x,u,v)}{\partial u}\|_{L^{\beta}(\Omega)}\\
&\quad + \|\frac{\partial F(x,u_{n},v_{n})}{\partial v}-\frac{\partial F(x,u,v)}{\partial v}\|_{L^{\beta}(\Omega)})\|(a,b)\|
\end{align*}
and so
\begin{align*}
\|\psi'(u_{n},v_{n})-\psi'(u,v)\|
&\leq S^{-1/p}(\|\frac{\partial F(x,u_{n},v_{n})}{\partial
u}-\frac{\partial
F(x,u,v)}{\partial u}\|_{L^{\beta}(\Omega)}\\
&\quad +\|\frac{\partial F(x,u_{n},v_{n})}{\partial v}-\frac{\partial
F(x,u,v)}{\partial
v}\|_{L^{\beta}(\Omega)})
\to  0 \quad\text{as } n\ \to \infty.
\end{align*}
\end{proof}

 From the above lemmas, we have $J_{\lambda,\mu}\in C^{1}(E,R)$.

As the energy functional $J_{\lambda,\mu}$ is not bounded
below on $E$, it is useful to consider the functional on the
Nehari manifold
\[
N_{\lambda,\mu}=\{(u,v)\in E\backslash\{(0,0)\}|\langle
J'_{\lambda,\mu}(u,v),(u,v)\rangle=0\}.
\]
Thus, $(u,v)\in N_{\lambda,\mu}$ if and only if
\begin{equation}
\langle
J'_{\lambda,\mu}(u,v),(u,v)\rangle
=\|(u,v)\|^p-\int_{\Omega}F(x,u,v)dx -K_{\lambda,\mu}(u,v)=0.
\end{equation}
Note that $N_{\lambda,\mu}$ contains every nonzero solution of
problem \eqref{e1.1}. Moreover, we have the following results.

\begin{lemma} \label{lem2.4}
The energy functional $J_{\lambda,\mu}$ is coercive
and bounded below on $N_{\lambda,\mu}$.
\end{lemma}

\begin{proof}
 If $(u,v)\in N_{\lambda,\mu}$, then by the H\"older
inequality and the Sobolev imbedding theorem,
\begin{equation} \label{e2.4}
\begin{aligned}
J_{\lambda,\mu}(u,v)
& = \frac{p^{**}-p}{p^{**}p}\|(u,v)\|^p
 -\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u,v) \\
&\geq \frac{p^{**}-p}{p^{**}p}\|(u,v)\|^p
 -\frac{p^{**}-q}{p^{**}q}S^{-\frac{q}{p}}
|\Omega|^{\frac{p^{**}-q}{p^{**}}}(\lambda^{\frac{p}{p-q}}
 +\mu^{\frac{p}{p-q}})^{\frac{p-q}{p}}\|(u,v)\|^{q}.
\end{aligned}
\end{equation}
 Thus, $J_{\lambda,\mu}$ is coercive and bounded below on
$N_{\lambda,\mu}$.
\end{proof}

 Define $\Phi_{\lambda,\mu}(u,v)=\langle
J'_{\lambda,\mu}(u,v),(u,v)\rangle$.
Then for $(u,v)\in N_{\lambda,\mu}$,
\begin{align}
\langle\Phi'_{\lambda,\mu}(u,v),(u,v)\rangle
&= p\|(u,v)\|^p-p^{**}\int_{\Omega}F(x,u,v)dx
-qK_{\lambda,\mu}(u,v) \label{e2.5}\\
&= (p-p^{**})\int_{\Omega}F(x,u,v)dx-(q-p)K_{\lambda,\mu}(u,v) \label{e2.6}\\
&= (p-q)\|(u,v)\|^p-(p^{**}-q)\int_{\Omega}F(x,u,v)dx \label{e2.7}\\
&= (p-p^{**})\|(u,v)\|^p-(q-p^{**})K_{\lambda,\mu}(u,v). \label{e2.8}
\end{align}
Now, we split $N_{\lambda,\mu}$ into three parts:
\begin{gather*}
N_{\lambda,\mu}^{+}=\{(u,v)\in N_{\lambda,\mu}|\langle
\Phi'_{\lambda,\mu}(u,v),(u,v)\rangle>0\};\\
N_{\lambda,\mu}^{0}=\{(u,v)\in N_{\lambda,\mu}|\langle
\Phi'_{\lambda,\mu}(u,v),(u,v)\rangle=0\};\\
N_{\lambda,\mu}^{-}=\{(u,v)\in N_{\lambda,\mu}|\langle
\Phi'_{\lambda,\mu}(u,v),(u,v)\rangle<0\}.
\end{gather*}
Then, we have the following results.

\begin{lemma} \label{lem2.5}
Suppose that $(u_0,v_0)$  is a local
minimizer for $J_{\lambda,\mu}$ on $N_{\lambda,\mu}$ and that
 $(u_0,v_0)\not\in N_{\lambda,\mu}^{0}$.
Then $J'_{\lambda,\mu}(u_0,v_0)=0$ in $E^{-1}$
(the dual space of the Sobolev space $E$ ).
\end{lemma}

\begin{proof}
 If $(u_0,v_0)$ is a local minimizer for
$J_{\lambda,\mu}$ on $N_{\lambda,\mu}$, then $(u_0,v_0)$ is a
solution of the optimization problem minimize
$J_{\lambda,\mu}(u,v)$
subject to $\Phi_{\lambda,\mu}(u,v)=0$.
Hence, by the theory of Lagrange multiplies, there exists
$\theta\in R $, such that
\[
J'_{\lambda,\mu}(u_0,v_0)=\theta \;
\Phi'_{\lambda,\mu}(u_0,v_0)\quad \text{in } E^{-1}(\Omega),
\]
Thus,
\begin{equation} \label{e2.9}
\langle J'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}=\theta
\langle \Phi'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}.
\end{equation}
Since $(u_0,v_0)\in N_{\lambda,\mu}$, we have
$\langle J'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}=0$.
Moreover,\\
$\langle\Phi'_{\lambda,\mu}(u_0,v_0),(u_0,v_0)\rangle_{E}\neq0$,
by \eqref{e2.9}, $\theta=0$.
Thus, $J'_{\lambda,\mu}(u_0,v_0)=0$ in $ E^{-1}$
(the dual space of the Sobolev space $E$).
\end{proof}

\begin{lemma} \label{lem2.6}
 If
\[
0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C(p,q,N,K,S,|\Omega|),
\]
then $N_{\lambda,\mu}^{0}=\emptyset$.
\end{lemma}

\begin{proof}
Suppose otherwise, that is there exists $\lambda>0,\mu>0$ with
\[
0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C(p,q,N,K,S,|\Omega|)
\]
such that $N_{\lambda,\mu}^{0}\neq\emptyset$. Then for
$(u,v)\in N_{\lambda,\mu}^{0}$, by \eqref{e2.7}, \eqref{e2.8} we have
\begin{align*}
0&= \langle
\Phi'_{\lambda,\mu}(u,v),(u,v)\rangle
=(p-q)\|(u,v)\|^p-(p^{**}-q)\int_{\Omega}F(x,u,v)dx\\
&= (p-p^{**})\|(u,v)\|^p-(q-p^{**})K_{\lambda,\mu}(u,v).
\end{align*}
By the Minkowski inequality, the Sobolev imbedding theorem and
\eqref{e1.5},
\begin{align*}
\int_{\Omega}F(x,u,v)dx&\leq
K(\int_{\Omega}(|u|^p+|v|^p)^{\frac{p^{**}}{p}}dx)^{\frac{p}{p^{**}}
\cdot\frac{p^{**}}{p}}\\
&\leq K\Big(\Big(\int_{\Omega}|u|^{p^{**}}dx\Big)^{\frac{p}{p^{**}}}
+\Big(\int_{\Omega}|v|^{p^{**}}dx\Big)^{\frac{p}{p^{**}}}\Big)^{\frac{p^{**}}{p}}\\
&\leq KS^{-\frac{p^{**}}{p}}\Big(\int_{\Omega}|\Delta
u|^pdx+\int_{\Omega}|\Delta v|^pdx\Big)^{\frac{p^{**}}{p}}\\
&= KS^{-\frac{p^{**}}{p}}\|(u,v)\|^{p^{**}}.
\end{align*}
Thus,
\[
\|(u,v)\|\geq(\frac{p-q}{K(p^{**}-q)}S^{\frac{p^{**}}{p}})^{\frac{1}{p^{**}-p}}
\]
and
\[
\|(u,v)\|\leq(\frac{p^{**}-q}{p^{**}-p}S^{-\frac{q}{p}}|\Omega|^{\frac{p^{**}-q}{p^{**}}})^{\frac{1}{p-q}}
(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})^{\frac{1}{p}}.
\]
This implies
\[
\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}\geq
C(p,q,N,K,S,|\Omega|),
\]
which is a contradiction. Thus, we conclude that if
\[
0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C(p,q,N,K,S,|\Omega|),
\]
we have $N_{\lambda,\mu}^{0}=\emptyset$.
\end{proof}

By Lemma \ref{lem2.6}, we write
$N_{\lambda,\mu}=N_{\lambda,\mu}^{+}\cup N_{\lambda,\mu}^{-}$ and
define
\begin{gather*}
\theta_{\lambda,\mu} =\inf_{(u,v)\in N_{\lambda,\mu}}
J_{\lambda,\mu}(u,v)\\
\theta_{\lambda,\mu}^{+}=\inf_{(u,v)\in
N_{\lambda,\mu}^{+}}  J_{\lambda,\mu}(u,v); \\
\theta_{\lambda,\mu}^{-}=\inf_{(u,v)\in N_{\lambda,\mu}^{-}}
J_{\lambda,\mu}(u,v).
\end{gather*}
Then we have the following result.

\begin{lemma} \label{lem2.7}
\begin{itemize}
\item[(i)] If $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}
<C(p,q,N,K,S,|\Omega|)$,
then we have $\theta_{\lambda,\mu}\leq \theta_{\lambda,\mu}^{+}<0 $;

\item[(ii)] if $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0$, then
$\theta_{\lambda,\mu}^{-}>d_0$ for some constant
\[
d_0=d_0(p,q,N,K,S,|\Omega|,\lambda,\mu)>0.
\]
\end{itemize}
\end{lemma}

\begin{proof}
(i) Let $(u,v)\in N_{\lambda,\mu}^{+}$.  By \eqref{e2.7},
\[
\frac{p-q}{p^{**}-q}\|(u,v)\|^p>\int_{\Omega}F(x,u,v)dx
\]
and so
\begin{align*}
J_{\lambda,\mu}(u,v)
&= (\frac{1}{p}-\frac{1}{q})\|(u,v)\|^p
 +(\frac{1}{q}-\frac{1}{p^{**}})\int_{\Omega}F(x,u,v)dx\\
&< [(\frac{1}{p}-\frac{1}{q})
 +(\frac{1}{q}-\frac{1}{p^{**}})\frac{p-q}{p^{**}-q}]\|(u,v)\|^p\\
&= -\frac{2(p-q)}{qN}\|(u,v)\|^p<0.
\end{align*}
Thus, from the definition of $\theta_{\lambda,\mu}$ and
$\theta_{\lambda,\mu}^{+}$, we can deduce that
$\theta_{\lambda,\mu}\leq\theta_{\lambda,\mu}^{+}<0$.

(ii) Let $(u,v)\in N_{\lambda,\mu}^{-}$. By \eqref{e2.7},
\[
\frac{p-q}{p^{**}-q}\|(u,v)\|^p<\int_{\Omega}F(x,u,v)dx.
\]
Moreover, by the Minkowski inequality, the Sobolev imbedding theorem,
and \eqref{e1.5},
\begin{equation} \label{e2.10}
\int_{\Omega}F(x,u,v)dx\leq
KS^{-\frac{p^{**}}{p}}\|(u,v)\|^{p^{**}}.
\end{equation}
This implies
\begin{equation} \label{e2.11}
\|(u,v)\|>(\frac{p-q}{K(p^{**}-q)})^{\frac{1}{p^{**}-p}}
S^{\frac{N}{2p^2}}
 \quad \text{for all }  (u,v)\in N_{\lambda,\mu}^{-}.
\end{equation}
By \eqref{e2.4} in the proof of Lemma \ref{lem2.4}
\begin{align*}
J_{\lambda,\mu}(u,v)
&\geq \|(u,v)\|^{q}[\frac{p^{**}-p}{p^{**}p}\|(u,v)\|^{p-q}
-\frac{p^{**}-q}{p^{**}q}S^{-\frac{q}{p}}
|\Omega|^{\frac{p^{**}-q}{p^{**}}}(\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}})^{\frac{p-q}{p}}]\\
&>(\frac{p-q}{K(p^{**}-q)})^{\frac{q}{p^{**}-p}}
 S^{\frac{qN}{2p^2}}[\frac{p^{**}-p}{p^{**}p}
 S^{\frac{(p-q)N}{2p^2}}(\frac{p-q}{K(p^{**}-q)})
^{\frac{p-q}{p^{*}-p}}\\
&\quad -\frac{p^{**}-q}{p^{**}q}
 S^{-\frac{q}{p}}|\Omega|^{\frac{p^{**}-q}{p^{**}}}
 (\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})^{\frac{p-q}{p}}].
\end{align*}
Thus, if
$0<|\lambda|^{\frac{p}{p-q}}+|\mu|^{\frac{p}{p-q}}<C_0$,
then
\[
J_{\lambda,\mu}(u,v)>d_0 \quad \text{for all }  (u,v)\in
N_{\lambda,\mu}^{-},
\]
for some $d_0=d_0(p,q,N,K,S,|\Omega|,\lambda,\mu)>0$.
This completes the proof.
\end{proof}

 For each $(u,v)\in E$ with $\int_{\Omega}F(x,u,v)dx>0$,
set
\[
t_{\rm max}=(\frac{(p-q)\|(u,v)\|^p}{(p**-q)
\int_{\Omega}F(x,u,v)dx})^{\frac{1}{p**-p}}>0.
\]
 Then the following lemma holds, which is similar to the one
 in Brown and Wu \cite[Lemma 2.6]{b2}.

\begin{lemma} \label{lem2.8}
For each $(u,v)\in E$ with
$\int_{\Omega}F(x,u,v)dx>0$,  there are unique
$0<t^{+}<t_{\rm max}<t^{-}$ such that $(t^{+}u,t^{+}v)\in
N_{\lambda,\mu}^{+}, (t^{-}u,t^{-}v)\in N_{\lambda,\mu}^{-}$ and
\[
J_{\lambda,\mu}(t^{+}u,t^{+}v)=\inf_{0\leq t\leq
t_{\rm max}}J_{\lambda,\mu}(tu,tv);\quad
J_{\lambda,\mu}(t^{-}u,t^{-}v)=\sup_{t\geq0}J_{\lambda,\mu}(tu,tv).
\]
\end{lemma}

\section{Proof of Theorems \ref{thm1.1} and \ref{thm1.2}}

We will need  the following lemma.

\begin{lemma} \label{lem3.1}
\begin{itemize}
\item[(i)] If $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}
<C(p,q,N,K,S,|\Omega|)$,
then there exists a $(PS)_{\theta_{\lambda,\mu}}$-sequence
$\{(u_{n},v_{n})\}\subset N_{\lambda,\mu}$ in $E$ for
$J_{\lambda,\mu}$;

\item[(ii)] if $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0$,
then there exists a $(PS)_{\theta^{-}_{\lambda,\mu}}$-sequence
$\{(u_{n},v_{n})\}\subset N^{-}_{\lambda,\mu}$ in $E$ for
$J_{\lambda,\mu}$.
\end{itemize}
\end{lemma}

 The proof of the above lemma is almost the same as that
in Wu \cite{w3}; we omit it.

 First, we establish the existence of a local minimum for
$J_{\lambda,\mu}$ on $N_{\lambda,\mu}^{+}$.

\begin{theorem} \label{thm3.1}
If $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}
<C(p,q,N,K,S,|\Omega|)$
and {\rm (F1)-(F3)} hold, then $J_{\lambda,\mu}$ has a minimizer
$(u_0^{+},v_0^{+})$ in $N_{\lambda,\mu}^{+}$ and it satisfies
\begin{itemize}
\item[(i)]
$J_{\lambda,\mu}(u_0^{+},v_0^{+})
=\theta_{\lambda,\mu}=\theta_{\lambda,\mu}^{+}$;

\item[(ii)]
$ (u_0^{+},v_0^{+})$ is  a positive solution
of \eqref{e1.1}.
\end{itemize}
\end{theorem}

\begin{proof}
 By the Lemma \ref{lem3.1}(i), there exists a minimizing sequence
$\{(u_{n},v_{n})\}$ for $J_{\lambda,\mu}$ on $N_{_{\lambda,\mu}}$
such that
\begin{equation} \label{e3.1}
J_{\lambda,\mu}(u_{n},v_{n})=\theta_{\lambda,\mu}+o(1), \quad
J'_{\lambda,\mu}(u_{n},v_{n})=o(1) \quad \text{in }\ E^{-1}
\end{equation}
Then by Lemma \ref{lem2.4} and the compact imbedding theorem, there exist a
subsequence $\{(u_{n},v_{n})\}$ and $(u_0^{+},v_0^{+})\in E$
such that
\begin{equation} \label{e3.2}
\begin{gathered}
u_{n}\rightharpoonup u_0^{+} \quad \text{weakly in }
W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega), \\
 u_{n}\to u_0^{+} \quad \text{strongly in } L^{q}(\Omega) , \\
v_{n}\rightharpoonup v_0^{+}  \quad \text{weakly  in }
W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega),  \\
v_{n}\to v_0^{+}  \quad  \text{strongly  in } L^{q}(\Omega).
\end{gathered}
\end{equation}
This implies that $K_{\lambda,\mu}(u_{n},v_{n})\to
K_{\lambda,\mu}(u_0^{+},v_0^{+})$   as
$n\to\infty$.
By \eqref{e3.1} and \eqref{e3.2}, it is easy to prove that
$(u_0^{+},v_0^{+})$ is a weak solution of \eqref{e1.1}.
Since
\begin{align*}
J_{\lambda,\mu}(u_{n},v_{n})
&= \frac{2}{N}\|(u_{n},v_{n})\|^p
-\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_{n},v_{n})\\
&\geq -\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_{n},v_{n})
\end{align*}
and by Lemma \ref{lem2.7} (i),
\[
J_{\lambda,\mu}(u_{n},v_{n})\to\theta_{\lambda,\mu}<0 \quad
\text{as } n\to\infty.
\]
Letting $n\to\infty$, we see that
$K_{\lambda,\mu}(u_0^{+},v_0^{+})>0$.
Thus, $(u_0^{+},v_0^{+})$ is a nontrivial solution of \eqref{e1.1}.

 Now it follows that $u_{n}\to u_0^{+}$
strongly in $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$,
$v_{n}\to v_0^{+}$ strongly in $ W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)$ and
$J_{\lambda,\mu}(u_0^{+},v_0^{+})=\theta_{\lambda,\mu}$.
By $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}$ and applying
Fatou's lemma, we obtain
\begin{align*}
\theta_{\lambda,\mu}
&\leq J_{\lambda,\mu}(u_0^{+},v_0^{+})\\
&= \frac{2}{N}\|(u_0^{+},v_0^{+})\|^p
-\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_0^{+},v_0^{+})\\
&\leq \liminf_{n\to\infty}(\frac{2}{N}\|(u_{n},v_{n})\|^p
-\frac{p^{**}-q}{p^{**}q}K_{\lambda,\mu}(u_{n},v_{n}))\\
&\leq \liminf_{n\to\infty}J_{\lambda,\mu}(u_{n},v_{n})
=\theta_{\lambda,\mu}.
\end{align*}
This implies
\[
J_{\lambda,\mu}(u_0^{+},v_0^{+})=\theta_{\lambda,\mu},\quad
\lim_{n\to\infty}\|(u_{n},v_{n})\|^p=\|(u_0^{+},v_0^{+})\|^p.
\]
Let
$(\widetilde{u}_{n},\widetilde{v}_{n})
=(u_{n},v_{n})-(u_0^{+},v_0^{+})$,
then by Br\'ezis-Lieb lemma \cite{b1},
\[
\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p=\|(u_{n},v_{n})\|^p-\|(u_0^{+},v_0^{+})\|^p.
\]
Therefore, $u_{n}\to u_0^{+}$ strongly in
$W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$,
$v_{n}\to v_0^{+}$ strongly in
$W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$.
Moreover, we have $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}^{+}$.
In fact, if $(u_0^{+},v_0^{+})\in N_{\lambda,\mu}^{-}$, by
Lemma \ref{lem2.8}, there are unique $t_0^{+}$ and $t_0^{-}$ such that
$(t_0^{+}u_0^{+},t_0^{+}v_0^{+})\in N_{\lambda,\mu}^{+}$
and $(t_0^{-}u_0^{+},t_0^{-}v_0^{+})\in
N_{\lambda,\mu}^{-}$. In particular, we have
$t_0^{+}<t_0^{-}=1$. Since
\[
\frac{d}{dt}J_{\lambda,\mu}(t_0^{+}u_0^{+},t_0^{+}v_0^{+})=0
\quad \text{and}\quad
\frac{d^2}{dt^2}J_{\lambda,\mu}(t_0^{+}u_0^{+},t_0^{+}v_0^{+})>0,
\]
there exists $t_0^{+}<\overline{t}\leq t_0^{-}$ such that
$J_{\lambda,\mu}(t_0^{+}u_0^{+},t_0^{+}v_0^{+})
<J_{\lambda,\mu}(\overline{t}u_0^{+},\overline{t}v_0^{+})$.
By Lemma \ref{lem2.8},
\[
J_{\lambda,\mu}(t_0^{+}u_0^{+},t_0^{+}v_0^{+})
<J_{\lambda,\mu}(\overline{t}u_0^{+},\overline{t}v_0^{+})
\leq J_{\lambda,\mu}(t_0^{-}u_0^{+},t_0^{-}v_0^{+})
=J_{\lambda,\mu}(u_0^{+},v_0^{+}),
\]
which is a contradiction.
It follows from the maximum principle that $(u_0^{+},v_0^{+})$
is a positive solution of \eqref{e1.1}.
This completes the proof.
\end{proof}

 The following two lemmas are similar to those in Hsu \cite{h1}.

\begin{lemma} \label{lem3.2}
If $\{(u_{n},v_{n})\}\subset E$ is a
$(PS)_{c}$-sequence for $J_{\lambda,\mu}$ with
$(u_{n},v_{n})\rightharpoonup(u,v)$ in $E$, then
$J'_{\lambda,\mu}(u,v)=0$, and there exists a positive
constant $\Lambda$ depending on $p,q,N,S$ and $|\Omega|$, such
that
$J_{\lambda,\mu}(u,v)\geq-\Lambda(\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}})$.
\end{lemma}

\begin{lemma} \label{lem3.3}
 If $\{(u_{n},v_{n})\}\subset E$ is a
$(PS)_{c}$-sequence for $J_{\lambda,\mu}$, then
$\{(u_{n},v_{n})\}$ is bounded
in $E$.
\end{lemma}

 Define
\[
S_{F}:=\inf_{(u,v)\in E}\{\frac{\|(u,v)\|^p}{(\int_{\Omega}
F(x,u,v)dx)^{\frac{p}{p^{**}}}}:\int_{\Omega}F(x,u,v)dx>0\}.
\]
We need also the following version of Br\'ezis-Lieb lemma
\cite{b1}.

\begin{lemma} \label{lem3.4}
Consider $F\in C^{1}(\overline{\Omega},(\mathbb{R}^+)^2)$ with
$F(x,0,0)=0$ and
\[
|\frac{\partial F(x,u,v)}{\partial u}|,|\frac{\partial
F(x,u,v)}{\partial v}|\leq C_{1}(|u|^{p-1}+|v|^{p-1})
\]
for some $1\leq p<\infty, C_{1}>0$.
Let $(u_{k},v_{k})$ be a bounded
sequence in $L^p(\overline{\Omega},(\mathbb{R}^+)^2)$, and such that
$(u_{k},v_{k})\rightharpoonup(u,v)$ weakly in $E$. Then
as $k\to\infty$,
\[
\int_{\Omega}F(x,u_{k},v_{k})dx\to
\int_{\Omega}F(x,u_{k}-u,v_{k}-v)dx+\int_{\Omega}F(x,u,v)dx.
\]
\end{lemma}

\begin{lemma} \label{lem3.5}
$J_{\lambda,\mu}$ satisfies the $(PS)_{c}$
condition with $c$ satisfying
\[
-\infty<c<c_{\infty}=\frac{2}{N}S_{F}^{N/(2p)}
-\Lambda(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}).
\]
\end{lemma}

\begin{proof}
 Let $\{(u_{n},v_{n})\}\subset E$ be a $(PS)_{c}$-sequence for
$J_{\lambda,\mu}$ with $c\in (-\infty, c_{\infty})$.
It follows from Lemma \ref{lem3.3} that $\{(u_{n},v_{n})\}$ is bounded in
$E$, and then $(u_{n},v_{n})\rightharpoonup(u,v)$ up to a
subsequence, $(u,v)$ is a critical point of $J_{\lambda,\mu}$.
Furthermore, we may assume
\begin{gather*}
u_{n} \rightharpoonup u,\quad v_{n}\rightharpoonup v\quad  \text{in }
W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega),\\
u_{n}\to u,\quad v_{n}\to v \quad \text{in } L^{q}(\Omega),\\
u_{n}\to u,\quad v_{n}\to v\quad  \text{a.e. on } \Omega.
\end{gather*}
Hence we have $J'_{\lambda,\mu}(u,v)=0$ and
\begin{equation} \label{e3.3}
\int_{\Omega}(\lambda|u_{n}|^{q}+\mu|v_{n}|^{q})dx\to
\int_{\Omega}(\lambda|u|^{q}+\mu|v|^{q})dx.
\end{equation}
Let $\widetilde{u}_{n}=u_{n}-u,\widetilde{v}_{n}=v_{n}-v$. Then by
Br\'ezis-Lieb lemma \cite{b1},
\begin{equation} \label{e3.4}
\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p\to\|(u_{n},v_{n})\|^p
-\|(u,v)\|^p\quad\text{as } n\to\infty.
\end{equation}
and by Lemma \ref{lem3.4},
\begin{equation} \label{e3.5}
\int_{\Omega}F(x,\widetilde{u}_{n},\widetilde{v}_{n})dx\to
\int_{\Omega}F(x,u_{n},v_{n})dx-\int_{\Omega}F(x,u,v)dx.
\end{equation}
Since
$J_{\lambda,\mu}(u_{n},v_{n})=c+o(1),J'_{\lambda,\mu}(u_{n},v_{n})=o(1)$
and \eqref{e3.3}-\eqref{e3.5}, we  deduce that
\begin{equation} \label{e3.6}
\frac{1}{p}\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p-\frac{1}{p^{**}}
\int_{\Omega}F(x,\widetilde{u}_{n},
\widetilde{v}_{n})dx=c-J_{\lambda,\mu}(u,v)+o(1).
\end{equation}
and
\[
\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p
-\int_{\Omega}F(x,\widetilde{u}_{n},\widetilde{v}_{n})dx=o(1).
\]
Hence, we may assume that
\begin{equation} \label{e3.7}
\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p\to l,\quad
\int_{\Omega}F(x,\widetilde{u}_{n},\widetilde{v}_{n})dx\to l.
\end{equation}
If $l=0$, the proof is complete. Assume $l>0$, then
from \eqref{e3.7}, we obtain
\[
S_{F}l^{\frac{p}{p^{**}}}= S_{F}\lim_{n\to\infty}
(\int_{\Omega}F(x,\widetilde{u}_{n},\widetilde{v}_{n})dx)
^{p/p^{**}}
\leq \lim_{n\to\infty}\|(\widetilde{u}_{n},\widetilde{v}_{n})\|^p=l,
\]
which implies  $l\geq S_{F}^{N/(2p)}$.
In addition, from Lemma \ref{lem3.2}, \eqref{e3.6} and \eqref{e3.7}, we obtain
\[
c= (\frac{1}{p}-\frac{1}{p^{**}})l+J_{\lambda,\mu}(u,v)
\geq \frac{2}{N}S_{F}^{N/(2p)}
-\Lambda(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}),
\]
which contradicts
$c<\frac{2}{N}S_{F}^{N/(2p)}-\Lambda(\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}})$.
\end{proof}

\begin{lemma} \label{lem3.6}
 There exist a nonnegative function $(u,v)\in
E\backslash\{(0,0)\}$ and $C^{*}>0$ such that for
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$, we have
\[
\sup_{t\geq0}J_{\lambda,\mu}(tu,tv)<c_{\infty}.
\]
In particular, $\theta^{-}_{\lambda,\mu}<c_{\infty}$ for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$.
\end{lemma}

\begin{proof}
Since $x_0\in \Omega$, there is $\rho_0>0$ such
that $B^{N}(x_0;2\rho_0)\subset\Omega$. Now, we consider the
functional $I:E\to R$ defined by
\[
I(u,v)=\frac{1}{p}\|(u,v)\|^p-\frac{1}{p^{**}}
\int_{\Omega}F(x,u,v)dx
\]
and define a cut-off function $\eta(x)\in C_0^{\infty}(\Omega)$
such that $\eta(x)=1$ for $|x-x_0|<\rho_0,\eta(x)=0$ for
$|x-x_0|>2\rho_0, 0\leq\eta\leq1$ and $|\nabla \eta|\leq C$.
For $\varepsilon>0$, let
\[
u_{\varepsilon}(x)=\eta(x)U(\frac{x}{\varepsilon}),
\]
where $U(\cdot)$ is a radially symmetric minimizer of
$\{\frac{\|\Delta u\|^p_{L^p}}{\|u\|^p_{L^{p^{**}}}}\}_{u\in
W^{2,p}(\mathbb{R}^N)\backslash\{0\}}$.
Similar to the work of Brown and Wu \cite{b3}, we have the following
estimates:
\begin{equation} \label{e3.8}
\begin{gathered}
\Big(\int_{\Omega}|u_{\varepsilon}|^{p^{**}}dx\Big)^{\frac{p}{p^{**}}}=
\varepsilon^{-\frac{N-2p}{p}}\|U\|_{L^{p^{**}}(\mathbb{R}^N)}^p
+O(\varepsilon),\\
\int_{\Omega}|\Delta
u_{\varepsilon}|^pdx=\varepsilon^{-\frac{N-2p}{p}}\|\Delta
U\|_{L^p(\mathbb{R}^N)}^p+O(1), \\
\frac{\int_{\Omega}|\Delta
u_{\varepsilon}|^pdx}{(\int_{\Omega}|u_{\varepsilon}|^{p^{**}}dx)^{\frac{p}{p^{**}}}}=
S+O(\varepsilon^{\frac{N-2p}{p}}),
\end{gathered}
\end{equation}
 Thus, we obtain
\[
\frac{\|\Delta
U\|^p_{L^p(\mathbb{R}^N)}}{\|U\|^p_{L^{p^{**}}(\mathbb{R}^N)}}=S=\inf_{u\in
W^{2,p}(\mathbb{R}^N)\backslash\{0\}}\frac{\|\Delta
u\|^p_{L^p(\mathbb{R}^N)}}{\|u\|^p_{L^{p^{**}}(\mathbb{R}^N)}}.
\]
Set $u_0(x)=e_{1}u_{\varepsilon}(x-x_0),
v_0(x)=e_2u_{\varepsilon}(x-x_0)$ and $(u_0,v_0)\in E$,
where $x_0\in \Omega$, $(e_{1},e_2)\in (\mathbb{R}^+)^2$, and $
e^p_{1}+e^p_2=1$ are such that
\[
F(x_0,e_{1},e_2)=\max_{x\in\overline{\Omega},g_{1}^p
+g_2^p=1,g_{1},g_2>0}F(x,g_{1},g_2)=:K .
\]
 Then, by (F1), \eqref{e1.5}, the definition of $S_{F}$ and
\eqref{e3.8}, we obtain
\begin{equation} \label{e3.9}
\begin{aligned}
\sup_{t\geq0}I(tu_0,tu_0)
&\leq   \frac{2}{N} (\frac{(e^p_{1}+e^p_2)\int_{\Omega}|\Delta
u_{\varepsilon}|^pdx}{(\int_{\Omega}F(x,e_{1}u_{\varepsilon}(x-x_0),
e_2u_{\varepsilon}(x-x_0))dx)^{\frac{p}{p^{**}}}})^{N/(2p)}\\
&=  \frac{2}{N}(\frac{\int_{\Omega}|\Delta
u_{\varepsilon}|^pdx}{\int_{\Omega}(|u_{\varepsilon}(x-x_0)|^{p^{**}}
F(x,e_{1},e_2)dx)^{\frac{p}{p^{**}}}})^{N/(2p)}\\
&\leq  \frac{2}{N}(\frac{1}{K^{\frac{p}{p^{**}}}})^{N/(2p)}(S+O(\varepsilon^{\frac{N-2p}{p}}))^{N/(2p)}\\
&=  \frac{2}{N}(\frac{1}{K^{\frac{p}{p^{**}}}})^{N/(2p)}
(S^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}}))\\
&\leq \frac{2}{N}S_{F}^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}}),
\end{aligned}
\end{equation}
where we have used that
\[
\sup_{t\geq0}(\frac{t^p}{p}A-\frac{t^{p^{**}}}{p^{**}}B)
=\frac{2}{N}(\frac{A}{B^{\frac{p}{p^{**}}}})^{N/(2p)},\quad
A,B>0.
\]
We can choose $\delta_{1}>0$ such that for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<\delta_{1}$, so we
have
\[
c_{\infty}=\frac{2}{N}S_{F}^{N/(2p)}-\Lambda(\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}})>0.
\]
Using the definitions of $J_{\lambda,\mu}$ and $(u_0,v_0)$, we obtain
\[
J_{\lambda,\mu}(tu_0,tv_0)\leq\frac{t^p}{p}\|(u_0,v_0)\|^p
\quad \text{for all }t\geq0,\; \lambda,\mu>0,
\]
which implies that there exists $t_0\in (0,1)$ satisfying
\[
\sup_{0\leq t\leq
t_0}J_{\lambda,\mu}(t_0u_0,t_0v_0)<c_{\infty},
\quad \text{for all }0<\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}}<\delta_{1}.
\]
Using the definitions of $J_{\lambda,\mu}$ and $(u_0,v_0)$, we
obtain
\begin{equation} \label{e3.10}
\begin{aligned}
\sup_{t\geq t_0}J_{\lambda,\mu}(tu_0,tv_0)
&=\sup_{t\geq t_0}(I_{\lambda,\mu}(tu_0,tv_0)-\frac{t^{q}}{q}
 K_{\lambda,\mu}(u_0,v_0))\\
&\leq \frac{2}{N}S_{F}^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}})
 -\frac{t_0^{q}}{q}
(e_{1}^{q}\lambda+e_2^{q}\mu)\int_{B^{N}(0;\rho_0)}|u_{\varepsilon}|^{q}dx\\
&\leq \frac{2}{N}S_{F}^{N/(2p)}+O(\varepsilon^{\frac{N-2p}{p}})
 -\frac{t_0^{q}}{q}m
(\lambda+\mu)\int_{B^{N}(0;\rho_0)}|u_{\varepsilon}|^{q}dx,
\end{aligned}
\end{equation}
where $m=\min\{e_{1}^{q},e_2^{q}\}$.
Let $0<\varepsilon\leq\rho_0^{\frac{p}{p-1}}$, we obtain
\begin{align*} %3.11
\int_{B^{N}(0;\rho_0)}|u_{\varepsilon}|^{q}dx
&=\int_{B^{N}(0;\rho_0)}\frac{1}{(\varepsilon+|x|^{\frac{p}{p-1}})
^{\frac{N-2p}{p}q}}dx\\
&\geq \int_{B^{N}(0;\rho_0)}\frac{1}{(2\rho_0^{\frac{p}{p-1}}
)^{\frac{N-2p}{p}q}}dx
= C_2(N,p,q,\rho_0).
\end{align*}
Combining with \eqref{e3.10} and the above inequality,  for all
$\varepsilon=(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})^{\frac{p}{N-2p}}\in
(0,\rho_0^{\frac{p}{p-1}})$, we have
\begin{equation} \label{e3.12}
\sup_{t\geq
t_0}J_{\lambda,\mu}(tu_0,tv_0)\leq\frac{2}{N}S_{F}^{N/(2p)}+
O(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})-\frac{t_0^{q}}{q}mC_2(\lambda+\mu).
\end{equation}
Hence, we can choose $\delta_2>0$ such that for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<\delta_2$, we
obtain
\[
O(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}})-\frac{t_0^{q}}{q}mC_2(\lambda+\mu)
<-\Lambda(\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}).
\]
If we set
$C^{*}=\min\{\delta_{1},\rho_0^{\frac{N-2p}{p-1}},\delta_2\}$
and
$\varepsilon=(\lambda^{\frac{p}{p-q}}
+\mu^{\frac{p}{p-q}})^{\frac{p}{N-2p}}$
then for $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$, we
have
\begin{equation}
\sup_{t\geq t_0}J_{\lambda,\mu}(tu_0,tv_0)<c_{\infty}.
\end{equation}
Finally, we prove that $\theta^{-}_{\lambda,\mu}<c_{\infty}$ for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$. Recall that
$(u_0,v_0)=(e_{1}u_{\varepsilon},e_2u_{\varepsilon})$. It is
easy to see that
\[
\int_{\Omega}F(x,u_0,v_0)dx>0.
\]
Combining this with Lemma \ref{lem2.8}, from the definition of
$\theta^{-}_{\lambda,\mu}$ and \eqref{e3.12},
we obtain that there exists $t_0>0$ such that
$(t_0u_0,t_0v_0)\in N_{\lambda,\mu}^{-}$ and
\[
\theta^{-}_{\lambda,\mu}\leq
J_{\lambda,\mu}(t_0u_0,t_0v_0)\leq\sup_{t\geq
0}J_{\lambda,\mu}(tu_0,tv_0)<c_{\infty}
\]
for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$.
\end{proof}

\begin{theorem} \label{thm3.2}
 If $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0^{*}$ and
{\rm(F1)--(F3)} hold, then $J_{\lambda,\mu}$ has a minimizer
$(u_0^{-},v_0^{-})$ in $N_{\lambda,\mu}^{-}$ and it satisfies
\begin{itemize}
\item[(i)] $J_{\lambda,\mu}(u_0^{-},v_0^{-})=\theta_{\lambda,\mu}^{-}$;

\item[(ii)] $(u_0^{-},v_0^{-})$  is  a positive  solution of
 \eqref{e1.1}.
\end{itemize}
where $C_0^{*}=\min\{C^{*},C_0\}$.
\end{theorem}

\begin{proof}
 By lemma \ref{lem3.1} (ii), there is a
$(PS)_{\theta^{-}_{\lambda,\mu}}$-sequence
$\{(u_{n},v_{n})\}\subset N^{-}_{\lambda,\mu}$ in $E$ for
$J_{\lambda,\mu}$ for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0$.
From Lemmas \ref{lem3.5}, \ref{lem3.6} and \ref{lem2.7} (ii), for
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C^{*}$,
$J_{\lambda,\mu}$ satisfies $(PS)_{\theta^{-}_{\lambda,\mu}}$
condition and $\theta^{-}_{\lambda,\mu}>0$. Since
$J_{\lambda,\mu}$ is coercive on $N_{\lambda,\mu}$, we obtain that
$(u_{n},v_{n})$ is bounded in $E$. Therefore, there exist a
subsequence still denoted by $(u_{n},v_{n})$ and
$(u_0^{-},v_0^{-})\in N_{\lambda,\mu}^{-}$ such that
$(u_{n},v_{n})\to(u_0^{-},v_0^{-})$ strongly in $E$
and
$J_{\lambda,\mu}(u_0^{-},v_0^{-})=\theta^{-}_{\lambda,\mu}>0$
for all $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0^{*}$.
Finally, by the same arguments as in the proof of
Theorem \ref{thm3.1}, for
all $0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C_0^{*}$, we
have that $(u_0^{-},v_0^{-})$ is a positive solution of
\eqref{e1.1}.
\end{proof}

 Now, we complete the proof of Theorems \ref{thm1.1} and \ref{thm1.2}.
By Theorem \ref{thm3.1}, we obtain that for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}<C(p,q,N,K,S,|\Omega|)$,
problem \eqref{e1.1} has a positive solution
$(u_0^{+} ,v_0^{+} )\in N_{\lambda , \mu}^{+}$.
On the other hand, from Theorem \ref{thm3.2},
we obtain the second positive solution
$(u_0^{-} ,v_0^{-} )\in N_{\lambda , \mu}^{-}$ for all
$0<\lambda^{\frac{p}{p-q}}+\mu^{\frac{p}{p-q}}
<C_0^{*}<C(p,q,N,K,S,|\Omega|)$.
Since $N_{\lambda , \mu}^{+} \cap N_{\lambda ,
\mu}^{-}=\emptyset$, this implies that
 $(u_0^{+},v_0^{+})$ and $(u_0^{-},v_0^{-})$ are distinct.
This completes the proof of Theorems \ref{thm1.1} and \ref{thm1.2}.


\subsection*{Acknowledgments}
This research was supported by grant 10871096 from the NNSF of China.
The authors would like to thank the anonymous referees for
their many valuable comments and suggestions which improved this
article.

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\end{document}