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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 71, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/71\hfil Positive solutions]
{Existence of positive solutions for quasilinear elliptic systems
involving the $p$-Laplacian}

\author[X. Shang, J. Zhang\hfil EJDE-2009/71\hfilneg]
{Xudong Shang, Jihui Zhang}  % in alphabetical order

\address{Xudong Shang \newline
Department of Mathematics, Nanjing Normal University Taizhou
College, Taizhou 225300, Jiangsu, China}
\email{xudong-shang@163.com}

\address{Jihui Zhang \newline
Institute of Mathematics, School of Mathematics and Computer
Sciences, Nanjing Normal University, Nanjing 210097, Jiangsu, China}
\email{jihuiz@jlonline.com}

\thanks{Submitted March 6, 2009. Published June 1, 2009.}
\subjclass[2000]{34A34, 34B18}
\keywords{$p$-Laplacian system; positive solutions; operator of
type $(S)_+$}

\begin{abstract}
 In this article, we study the existence of positive solutions for
 the quasilinear elliptic system
 \begin{gather*}
 -\Delta_{p}u = f(x,u,v) \quad x \in \Omega ,\\
 -\Delta_{p}v = g(x,u,v) \quad x \in \Omega ,\\
 u = v = 0 \quad x \in \partial\Omega.
 \end{gather*}
 Using degree theoretic arguments based on the degree map for
 operators of type $(S)_{+}$, under suitable assumptions on the
 nonlinearities,  we prove the existence of positive weak solutions.
\end{abstract}

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

\section{Introduction and main result}

In this paper we study the existence of positive solution for the
 nonlinear elliptic system
\begin{equation}\label{e1.1}
\begin{gathered}
-\Delta_{p}u = f(x,u,v) \quad x \in \Omega ,\\
-\Delta_{p}v = g(x,u,v) \quad x \in \Omega ,\\
u = v = 0 \quad x \in \partial\Omega
\end{gathered}
\end{equation}
 where $\Delta_{p}u = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian operator, and $\Omega$ is a smooth bounded region in
$\mathbb{R}^N$ for $N\geq 1$,  $p> 1$.

Elliptic systems have several practical applications. For example
they can describe the multiplicative chemical reaction catalyzed by
 grains under constant or variant temperature, a
correspondence of the stable station of dynamical system determined
by the reaction-diffusion system. In recent years, many publications
have appeared concerning quasilinear elliptic systems which have
been used in a great variety of applications, we refer the readers
to \cite{zhang04, Djellit07, Djellit04, Drabek03, Hai07, Djellit03}
and the references therein. Existence and multiplity results for
quasilinear elliptic systems with variational have been broadly
investigate.  Djellit and Tas  \cite{Djellit07} studied a class of
quasilinear elliptic systems involving the $p$-Laplacian operator,
the right hand sides of systems being closed related to the critical
Sobolev exponents. Under some additional assumptions on the
nonlinearities, they proved the existence of at least one nontrivial
solution. In \cite{ Drabek03}, the authors studied the existence and
multiplicity of non-negative solutions for the quasilinear elliptic
system in both bounded and unbounded domain in $\mathbb{R}^N$.
Zhang \cite{zhang04} used the Leray-Schauder degree to obtain a
positive solution of the nonlinear elliptic system.

In this work, we show the existence of positive solutions for system
\eqref{e1.1}.
 Using the degree theory for $(S)_{+}$ operator
initiated by Browder \cite{Browder83}. Our main goal in this article
is to extend the main result of \cite{zhang04} to the quasilinear
case. The relevant studies
about $(S)_{+}$ operators can be found in \cite{Lu05, Aman82, Zeidler90}.

 Through this paper, $(u,v) \in \mathbb{R}^2$. As to the
 nonlinearities $f,g$, we assume that they are Caratheodory
 functions satisfying the following growth conditions:

\noindent (i) There exist $ a_{i}\geq 0, c_{i} \geq 0$$(i=1,2)$
such that
\begin{gather*}
0\leq f(x,u,v) \leq a_{1}|(u,v)|^{q-1} + c_{1},
\\
0\leq g(x,u,v) \leq a_{2}|(u,v)|^{q-1} + c_{2}
\end{gather*}
where $1<p < q < p^{*} = \frac{Np}{N-p}$ if $p < N$, or $p < q <
+\infty$ if $p \geq N$.

\noindent (ii) There existence an $\epsilon^{'}>0$, $c_{3}>0$,
$1<p<\theta<p^{*}$ such that
\[
f(x,u,v)u + g(x,u,v)v\leq
(\lambda_{1}-\epsilon')(|u|^{p}+|v|^{p})+c_{3}(|u|^{\theta}+|v|^{\theta})
\]
where $\lambda_{1}$ stands for the first eigenvalue of the operator
$-\Delta_{p}$ in $W_{0}^{1,p}(\Omega)$.

\noindent(iii)  $f(x,u,v)$ and $g(x,u,v)$ also satisfies
\[
\liminf_{|(u,v)|\to \infty}\frac{f(x,u,v)}{|(u,v)|^{p-1}} =
+\infty,\quad
\liminf_{|(u,v)|\to \infty}\frac{g(x,u,v)}{|(u,v)|^{p-1}} = +\infty.
\]

 The main result of this paper is as follows.

\begin{theorem} \label{thm1.1}
 Suppose that {\rm (i)--(iii)}  hold.
Then  \eqref{e1.1} has a positive weak solution.
\end{theorem}


 The plan of this paper is as follows. In section 2, we shall
present some lemmas in order to prove our main results. The main
results is proved in section 3.


\section{Preliminaries}

We  start this section by recalling the definition for
operator of type $(S)_{+}$.

\begin{definition}[\cite{Zeidler90}] \label{def2.1} \rm
 Let $X$ be a reflexive
Banach space and $X^{*}$ its topological dual. A mapping
$A: X\to X^{*}$ is of type $(S)_{+}$, if for each sequence
${u_{n}}$ in $X$ satisfying $u_{n}\rightharpoonup u_{0}$ in $X$
and
$$
\limsup_{n\to +\infty} \langle A(u_{n}), u_{n}-u_{0} \rangle \leq 0,
$$
we have  $u_{n}\to u_{0}$.
\end{definition}

 If the operator $A$ satisfies the above condition, then it is
possible to define its degree.
 Now we consider triples $(A,\Omega,x_{0})$ such that $\Omega$
 is a nonempty, bounded, open set in $X$,
$A: \overline{\Omega}\to  X^{*}$ is a demicontinuous mapping of
type $(S)_{+}$ and $x_{0}\not\in A(\partial  \Omega)$.
On such triples Browder \cite{Browder83} defined a degree denoted by
 $\deg (A,\Omega,x_{0})$, which has the following three basic
 properties:
\begin{itemize}
\item[(i)]  (Normality) If $x_{0}\in  A(\Omega)$ then
$\deg (A,\Omega,x_{0}) = 1$;

\item[(ii)] (Domain additivity) If
 $\Omega_{1},\Omega_{2}$ are disjoint open subsets of $\Omega$ and
  $x_{0}\not\in  A(\overline{\Omega}\backslash(\Omega_{1}\cup\Omega_{2}))$
then
 $\deg (A,\Omega,x_{0}) = \deg (A,\Omega_{1},x_{0}) +
 \deg (A,\Omega_{2},x_{0})$;

\item[(iii)] (Homotopy invariance) If $\{A_{t}\}_{t\in [0,1]}$ is a
homotopy of type $(S)_{+}$ such that ${A_{t}}$ is bounded for every
$t\in [0,1]$ and $x_{0}:[0,1]\to X^{*}$ is a continuous map
such that $x_{0}(t)\not\in A_{t}(\partial\Omega)$ for all $t\in
[0,1]$, then $\deg (A_{t},\Omega,x_{0}(t)) $ is independent of
$t\in [0,1]$.
\end{itemize}

\begin{remark} \label{remark2.2}\rm
If the operator $A$ is of type $(S)_{+}$, and $K$ is compact, then
$A+K$ is of type $(S)_{+}$.
\end{remark}

\begin{lemma} \label{lemma2.3}
Assume $A$ is of type $(S)_{+}$. Suppose that for  $u\in X$
and $\|u\|_{X} = r$.  $\langle Au,u\rangle > 0$ is satisfied.
Then
$$
\deg (A,B_{r}(0),0) = 1.
$$
\end{lemma}

  In this paper, we denote by $Z$ the product space
$W_{0}^{1,p}(\Omega)\times W_{0}^{1,p}(\Omega)$.
The space $Z$ will be endowed with the norm
\[
\|z\|_{Z}^{p} =
\|u\|_{W_{0}^{1,p}(\Omega)}^{p}+\|v\|_{W_{0}^{1,p}(\Omega)}^{p},
\hskip 0.5cm z=(u,v)\in Z,
\]
where $\|u\|_{W_{0}^{1,p}(\Omega)} = (\int_{\Omega}|\nabla
 u|^{p}dx)^{1/p}$.

  As usual, a weak solution of system \eqref{e1.1} is any
$(u,v)\in Z$ such that
\[
\int_{\Omega} |\nabla u|^{p-2}\nabla u\nabla\xi dx + \int_{\Omega}
|\nabla v|^{p-2}\nabla v\nabla\zeta dx - \int_{\Omega} f(x,u,v)\xi
dx - \int_{\Omega} g(x,u,v)\zeta dx = 0
\]
for every $(\xi,\zeta)\in Z$.

  Next let us introduce the functionals
$I_{i},F_{i}: Z\to Z^{*}$ $(i=1,2)$ as follows:
\begin{gather*}
\langle I_{1}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} |\nabla
u|^{p-2}\nabla u\nabla\xi dx,
\\
\langle I_{2}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} |\nabla
v|^{p-2}\nabla v\nabla\zeta dx,
\\
\langle F_{1}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} f(x,u,v)\xi
dx ,
\\
\langle F_{2}(u,v),(\xi,\zeta) \rangle = \int_{\Omega} g(x,u,v)\zeta
dx .
\end{gather*}
Define the operator
\[
 A = I_{1} + I_{2} -F_{1}- F_{2}.
\]


\begin{lemma} \label{lemma2.4}
The mapping $B = I_{1} + I_{2}$ is of type $(S)_{+}$.
\end{lemma}

\begin{proof}
Assume that $(u_{n},v_{n})\rightharpoonup
(u_{0},v_{0})$ in $Z$ and
\[
\limsup_{n\to +\infty} \langle B(u_{n},v_{n}),
(u_{n}-u_{0},v_{n}-v_{0})\rangle \leq0.
\]
From the weak convergence we have that
\[
\limsup_{n\to +\infty} \langle B(u_{0},v_{0}),
(u_{n}-u_{0},v_{n}-v_{0})\rangle = 0.
\]
Then we get
\[
\limsup_{n\to +\infty} \langle B(u_{n},v_{n}) -
B(u_{0},v_{0}), (u_{n}-u_{0},v_{n}-v_{0})\rangle \leq 0.
\]
By the monotonicity property of $B$, we obtain
\[
\limsup_{n\to +\infty} \langle B(u_{n},v_{n}) -
B(u_{0},v_{0}), (u_{n}-u_{0},v_{n}-v_{0})\rangle = 0.
\]
This implies
\begin{displaymath}
\begin{aligned}
&\lim_{n\to +\infty} \int_{\Omega}(|\nabla u_{n}|^{p-2}\nabla
u_{n}- |\nabla u_{0}|^{p-2}\nabla u_{0})(\nabla u_{n}-\nabla
u_{0})dx
\\
&+ \int_{\Omega}(|\nabla v_{n}|^{p-2}\nabla v_{n}- |\nabla
v_{0}|^{p-2}\nabla v_{0})(\nabla v_{n}-\nabla v_{0})dx =  0.
\end{aligned}
\end{displaymath}
Observe  that for all $x,y\in \mathbb{R}^N$,
\[
|x-y|^{p}\leq \begin{cases}
(|x|^{p-2}x-|y|^{p-2}y)(x-y) &  \text{if } p\geq 2, \\
[(|x|^{p-2}x-|y|^{p-2}y)(x-y)]^{p/2}(|x|+|y|)^{(2-p)p/2}
&  \text{if }   1<p<2.
\end{cases}
\]
So, we obtain
\[
\lim_{n\to +\infty} \int_{\Omega}(|\nabla u_{n}-\nabla
u_{0}|^{p}+|\nabla v_{n}-\nabla v_{0}|^{p})dx = 0.
\]
Thus, we have $(\nabla u_{n},\nabla v_{n})\to (\nabla
u_{0},\nabla v_{0})$ in $L^{p}(\Omega)\times L^{p}(\Omega)$. Also
$(u_{n},v_{n})\rightharpoonup (u_{0},v_{0})$ in $Z$, which implies
that $(u_{n},v_{n})\to (u_{0},v_{0})$ in
$L^{p}(\Omega)\times L^{p}(\Omega)$. Hence
\[
 (u_{n},v_{n})\to (u_{0},v_{0}) \quad  \text{in }Z.
\]
The proof is complete.
\end{proof}

\begin{lemma} \label{lemma2.5}
 The mapping $F = F_{1}+F_{2}$ is  compact.
\end{lemma}

The proof of the above lemma follows easily from Hypotheses (i). By
Remark \ref{remark2.2} we have the following result.

\begin{lemma} \label{lemma2.6}
 The operator $A$ is type of $(S)_{+}$.
\end{lemma}


\section{Proof of main theorem}

 Define $B_{R}^{K} = \{(u,v)\in K:\|(u,v)\|_{Z} < R\}$,
$K =\{(u,v)\in K:u\geq 0,v\geq 0, \text{a.e.}x\in\Omega \} $.
Now, we give the proofs of the main results.

\begin{proof}[Proof of Theorem 1.1.]
There exists $R_{0}>0$ such that
\begin{equation}\label{e3.1}
\deg (A,B_{R}^{K},0) = 0  \quad \text{for all } R\geq R_{0}.
\end{equation}
 Let
\begin{equation}\label{e3.2}
\langle L(u,v),(\xi,\zeta)\rangle = \int_{\Omega}((k+\epsilon)
u^{p-1}\xi + (k+\epsilon) v^{p-1}\zeta) dx,
\quad \forall (\xi,\zeta) \in Z
\end{equation}
where $k$ is a real number, $0<\epsilon<\epsilon^{'}$. Since $L$ is
a completely continuous operator, the homotopy $H_{t}:[0,1]\times
K\to Z^{*}$ defined by
\begin{align*}
\langle H_{t}(u,v),(\xi,\zeta) \rangle
&= \int_{\Omega} (|\nabla u|^{p-2}\nabla u \nabla\xi
 + |\nabla v|^{p-2}\nabla v \nabla\zeta) dx \\
&\quad - (1-t)\int_{\Omega} (f(x,u,v)\xi + g(x,u,v)\zeta )dx \\
&\quad - t \int_{\Omega} ((k+\epsilon)u^{p-1}\xi
 + (k+\epsilon)v^{p-1}\zeta) dx,
\end{align*}
where the value of $k$ will be fixed later.
Clearly $H_{t}$ is of type $(S)_{+}$. We claim that there
exists $R_{0} > 0$ such that
\[
H_{t}(u,v)\not= 0 \quad \text{for all }
 t\in[0,1],\; (u,v)\in\partial B_{R}^{K} \; R\geq R_{0}.
\]
Suppose that is not true. Then we can find sequences
$\{t_{n}\}\subset [0,1]$ and $\{(u_{n},v_{n})\}\subset Z$ such that
$t_{n}\to t\in [0,1]$, $\|(u_{n},v_{n})\|_{Z}\to
\infty$ and
\begin{equation}\label{e3.3}
\begin{aligned}
&\int_{\Omega} (|\nabla u_{n}|^{p-2}\nabla u_{n} \nabla\xi
+ |\nabla v_{n}|^{p-2}\nabla v_{n} \nabla\zeta) dx \\
&= t_{n} \int_{\Omega}
((k+\epsilon)u_{n}^{p-1}\xi + (k+\epsilon)v_{n}^{p-1}\zeta) dx
\\
&\quad + (1-t_{n})\int_{\Omega} (f(x,u_{n},v_{n})\xi +
g(x,u_{n},v_{n})\zeta )dx
\end{aligned}
\end{equation}
for every $(\xi,\zeta) \in  Z$. Put $(\omega_{n},\psi_{n}) =
\frac{(u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}}$, we may assume that
there exists $(\omega_{0},\psi_{0})\in Z$ satisfying
\begin{gather*}
(\omega_{n},\psi_{n})\rightharpoonup (\omega_{0},\psi_{0})
      \quad \text{in } Z,\\
(\omega_{n},\psi_{n})\to (\omega_{0},\psi_{0}) \quad
\text{in } L^{p}(\Omega)\times L^{p}(\Omega), \\
(\omega_{n},\psi_{n})\to (\omega_{0},\psi_{0})
\quad \text{a.e. } x\in \Omega.
\end{gather*}
Applying the test function $(\omega_{n} - \omega_{0},\psi_{n} -
\psi_{0})\in Z $ in \eqref{e3.3}, we find
\begin{equation}\label{e3.4}
\begin{aligned}
&\int_{\Omega} (|\nabla
\omega_{n}|^{p-2}\nabla \omega_{n} \nabla(\omega_{n}- \omega_{0})+
|\nabla \psi_{n}|^{p-2}\nabla \psi_{n} \nabla(\psi_{n}- \psi_{0}))dx
\\
&=(1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\omega_{n}-
\omega_{0}) +
\frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\psi_{n}-
\psi_{0}))dx
\\
&\quad + t_{n} \int_{\Omega}
((k+\epsilon)\omega_{n}^{p-1}(\omega_{n}- \omega_{0}) +
(k+\epsilon)\psi_{n}^{p-1}(\psi_{n}- \psi_{0})) dx\,.
\end{aligned}
\end{equation}
We know that
\begin{gather*}
(1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n})
\|_{Z}^{p-1}}(\omega_{n}- \omega_{0}) +
\frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}(\psi_{n}-
\psi_{0}))dx \to 0  \quad n\to\infty,
\\
t_{n} \int_{\Omega} ((k+\epsilon)\omega_{n}^{p-1}(\omega_{n}-
\omega_{0}) + (k+\epsilon)\psi_{n}^{p-1}(\psi_{n}- \psi_{0}))
dx\to 0\quad n\to\infty.
\end{gather*}
By \eqref{e3.4}, we have
\[
\lim_{n\to +\infty} \int_{\Omega} (|\nabla
\omega_{n}|^{p-2}\nabla \omega_{n} \nabla(\omega_{n}- \omega_{0})+
|\nabla \psi_{n}|^{p-2}\nabla \psi_{n} \nabla(\psi_{n}- \psi_{0}))
dx = 0;
\]
i.e.,
\[
 \lim_{n\to +\infty} \langle
B(\omega_{n},\psi_{n}),
(\omega_{n},\psi_{n})-(\omega_{0},\psi_{0})\rangle = 0 .
\]
According to Lemma \ref{lemma2.4} we obtain
$(\omega_{n},\psi_{n})\to(\omega_{0},\psi_{0})$ in $Z$ as
$n\to\infty$ and $\|(\omega_{0},\psi_{0})\|=1$. This shows that
$(\omega_{0},\psi_{0})\not= (0,0)$. Let now apply the test function
$(\omega_{0},\psi_{0})\in Z$ in \eqref{e3.4}, we get
\begin{equation}\label{e3.5}
\begin{aligned}
&\int_{\Omega} (|\nabla
\omega_{n}|^{p-2}\nabla \omega_{n} \nabla \omega_{0}+ |\nabla
\psi_{n}|^{p-2}\nabla \psi_{n} \nabla\psi_{0}) dx
\\
&=(1-t_{n})\int_{\Omega}(\frac{f(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}
\omega_{0} + \frac{g(x,u_{n},v_{n})}{\|(u_{n},v_{n})\|_{Z}^{p-1}}
\psi_{0})dx
\\
&\quad + t_{n} \int_{\Omega} ((k+\epsilon)\omega_{n}^{p-1}\omega_{0} +
(k+\epsilon)\psi_{n}^{p-1}\psi_{0}) dx
\end{aligned}
\end{equation}
Passing to the limit as $n\to\infty$, using Fatou's lemma
and hypothesis $(\text{iii})$, we obtain
\begin{equation}\label{e3.6}
 \int_{\Omega} (|\nabla
\omega_{0}|^{p} + |\nabla \psi_{0}|^{p})dx \geq (k + \epsilon)
\int_{\Omega} (\omega_{0}^{p} + \psi_{0}^{p}) dx.
\end{equation}
Now we take $k=\frac{\|(\omega_{0},
\psi_{0})\|_{Z}^{p}}{\|(\omega_{0}, \psi_{0})\|_{L^{p}\times
L^{p}}^{p}}$, by \eqref{e3.6} we have
\[
 k+\epsilon \leq \frac{\int_{\Omega}(|\nabla
\omega_{0}|^{p} + |\nabla
\psi_{0}|^{p})dx}{\int_{\Omega}(\omega_{0}^{p} + \psi_{0}^{p})dx} =
 k.
\]
This contradiction shows the claim. Using the homotopy invariance of
the degree map, which through the homotopy $H_{t}$ yields
\[
\deg (A,B_{R}^{K},0) = \deg (H_{1},B_{R}^{K},0) \quad
\text{for all }  R\geq R_{0}.
\]
Now we computing $\deg (H_{1},B_{R}^{K},0)$. Let the homotopy
$H_{t}': [0,1]\times k\to Z^{*}$ be defined by
\begin{align*}
\langle H_{t}'(u,v),(\xi,\zeta)\rangle
&= \int_{\Omega} (|\nabla u|^{p-2}\nabla u \nabla\xi
+ |\nabla v|^{p-2}\nabla v \nabla\zeta) dx
\\
&\quad - t \int_{\Omega} (m(x)\xi + m(x)\zeta) dx
-\int_{\Omega}((k +\epsilon )u^{p-1}\xi + (k +\epsilon
)v^{p-1}\zeta) dx
\end{align*}
for all $(\xi,\zeta) \in Z$, $t\in [0,1]$,
$m(x)\in L_{+}^{\infty}(\Omega)=\{u(x)\in
L^{\infty}(\Omega)|u(x)\geq0,\forall x\in\Omega\}$.
Clearly, it is a
$(S)_{+}$ homotopy. So we have
\[
\deg (H_{1},B_{R}^{K},0) = \deg (H_{t}',B_{R}^{K},0).
\]
Similarly, we prove the claim concerning the homotopy $H_{t}'$.
By the homotopy invariance of the degree map, we have
\[
\deg (H_{1},B_{R}^{K},0) = \deg (H_{1}',B_{R}^{K},0).
\]

Next, we show that $\deg (H_{1}',B_{R}^{K},0)= 0$. If
$\deg (H_{1}',B_{R}^{K},0)\not= 0$, there exits
$(u,v)\in B_{R}^{K}$ such that
\begin{align*}
& \int_{\Omega} (|\nabla
u|^{p-2}\nabla u \nabla\xi + |\nabla v|^{p-2}\nabla v \nabla\zeta)
dx \\
&=  \int_{\Omega} (m(x)\xi + m(x)\zeta) dx
+\int_{\Omega}((k +\epsilon )u^{p-1}\xi + (k +\epsilon
)v^{p-1}\zeta) dx.
\end{align*}
Clearly $(u,v)\not = (0,0)$, let $(\xi,\zeta) = (u,v)$, then
\[
 \int_{\Omega} (|\nabla u|^{p} dx + |\nabla v|^{p} dx) \geq
(k+\epsilon)\int_{\Omega}(u^{p}+v^{p}) dx,
\]
We take $k=\frac{\|(u, v)\|_{Z}^{p}}{\|(u, v)\|_{L^{p}\times
L^{p}}^{p}}$,  which provides a contradiction. Therefore
\[
\deg (A,B_{R}^{K},0) = \deg (H_{1}',B_{R}^{K},0) =0.
\]
So \eqref{e3.1} holds.
Then, note that
\[
\lambda_{1} = \inf_{u\in W_{0}^{1,p}(\Omega)\backslash \{0\}}
\frac{\int_{\Omega}|\nabla u|^{p}dx}{\int_{\Omega}|u|^{p}dx},
\]
we have $\|u\|_{W_{0}^{1,p}(\Omega)}^{p} \geq
\lambda_{1}\|u\|_{L^{p}(\Omega)}^{p}$.
By(ii), we get
\begin{align*}
\langle A(u,v),(u,v)\rangle
&= \|(u,v)\|_{Z}^{p}-\int_{\Omega}(f(x,u,v)u + g(x,u,v)v) dx\\
&\geq \frac{\epsilon'}{\lambda_{1}}(\|u\|^{p}+\|v\|^{p})-
 c_{4}(\|u\|^{\theta}+\|v\|^{\theta}),
\end{align*}
where $c_{4} > 0$. Since $\theta > p$, there exist $r > 0$ such that
\[
\langle A(u,v),(u,v)\rangle > 0,
\]
for all $(u,v) \in \partial B_{r}^{K}$, where $B_{r}^{K} =
\{(u,v)\in K| \|(u,v)\|_{Z} < r\}$. In view of Lemma \ref{lemma2.3},
 there exists sufficiently small $r>0$ such that
\[
\deg (A,B_{r}^{K} ,0) = 1.
\]
According to the \eqref{e3.1}, we can take $R > r$ such that
\[
\deg (A,B_{R}^{K},0) = 0.
\]
Since the domain additivity of type $(S)_{+}$, we obtain
\[
\deg (A,B_{R}^{K}\backslash B_{r}^{K},0) = -1.
\]
So we are led to the existence of $(u,v)\in B_{r,R}^{K} = \{(u,v)\in
K| r<\|(u,v)\|_{Z} < R\}$ such that
$A(u,v) = 0$.
Hence, system\eqref{e1.1} has a positive solution.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referee for his/her
valuable suggestions.

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