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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 134, 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-2009/134\hfil Existence and uniqueness]
{Existence and uniqueness of solutions to a semilinear elliptic
system}

\author[Z. Chen, Y. Cui\hfil EJDE-2009/134\hfilneg]
{Zu-Chi Chen, Ying Cui} % in alphabetical order

\address{Zu-Chi Chen \newline
 Department of mathematics\\
 University of Science and Technology of China\\
 Hefei 230026, China}
\email{chenzc@ustc.edu.cn}

\address{Ying Cui \newline
 Department of mathematics\\
 University of Science and Technology of China\\
 Hefei 230026, China}
\email{cuiy@mail.ustc.edu.cn}

\thanks{Submitted July 6, 2009. Published October 19, 2009.}
\thanks{Supported by grant 10371116 from the NNSF of China}
\subjclass[2000]{35J55, 35J60, 35J65}
\keywords{Super-sub solutions; compact continuous operator;
\hfill\break\indent
 Leray-Schauder fixed point theorem}

\begin{abstract}
 In this article, we show the existence and uniqueness of smooth
 solutions for boundary-value problems of semilinear elliptic systems.
\end{abstract}

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

\section{Introduction and main results}

We study the solvability for the semilinear elliptic system with
homogeneous Dirichlet boundary value condition
\begin{equation} \label{e1.1}
\begin{gathered}
 L_1 u= f(x,u,v,Du,Dv),  \quad x\in \Omega \\
 L_2 v= g(x,u,v,Du,Dv),  \quad x\in \Omega \\
 u = v = 0, \quad x\in \partial\Omega
\end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ ($N\geq 2$) denotes a bounded
domain with smooth boundary, and $f,g: \overline{\Omega}\times
\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $L_1$ and $L_2$ are
the uniformly elliptic operators of second order:
$$
L_ku=\sum_{i,j=1}^N \partial_{x_{j}}(a_{i,j}^k(x)u), \ k=1,2,
$$
with its first eigenvalue $\lambda_k>0$ for $k=1,2,$ and in the
context, $\lambda=:\min\{\lambda_1,\lambda_2\}.$ \\
We suppose the following conditions:
\begin{itemize}
\item[(H1)] $f,g: \Omega \times \mathbb{R} \times \mathbb{R}
\times \mathbb{R}^N \times \mathbb{R}^N
\mapsto \mathbb{R}$ are Caratheodory functions which satisfy
\begin{gather*}
|f(x,s,t,\xi,\eta)| \leq h_1(x,s,t)+k_1|\xi|^{\alpha_1}+k_2|
 \eta|^{\alpha_2}, \\
|g(x,s,t,\xi,\eta)| \leq h_2(x,s,t)+k_3|\xi|^{\alpha_3}
+k_4|\eta|^{\alpha_4},
\end{gather*}
where constant $\alpha_i,k_i \in \mathbb{R}_0^+$, $i = 1,2,3,4$;
$h_1(x,s,t)$ and $h_2(x,s,t)$ are Caratheodory functions that
satisfy the following conditions: \item[(H2)]  for every $r>0$,
$\sup _{|s| \leq r,\, |t| \leq r} h_i(\cdot ,s,t) \in
L^p(\Omega)$, $\frac{2N}{N+1}<p<N$; \item[(H3)]  $\max
\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}
 =: \alpha \le 1$;
\item[(H4)] $\alpha_i \ge \frac{1}{p}$ or $\alpha_i = 0$, for
$i = 1,2,3,4$.
\end{itemize}

\begin{theorem} \label{thm1.1}
Assume {\rm (H1)--(H4)}. If  \eqref{e1.1} has two pairs of
subsolutions and supersolutions $(\underline u,\overline u),
(\underline v,\overline v)$, then \eqref{e1.1} has at least one
solution $(u, v)\in [W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)]^2$.
\end{theorem}

For the next theorem we need the assumption
\begin{itemize}
\item[(H5)] $f,g: \Omega \times \mathbb{R} \times \mathbb{R}
\times \mathbb{R}^N \times \mathbb{R}^N \mapsto \mathbb{R} $ are
Lipschitz continuous, with Lipschitz coefficients $l_1$ and $l_2$,
and $L:=\max\{l_1,l_2\}<\frac{\lambda}{4C+1}$, where
$C=C(n,p,\Omega)$ is the coefficient for the Poincar\'{e}
inequality.
\end{itemize}

\begin{theorem} \label{thm1.2}
Under Condition {\rm (H5)}, Problem \eqref{e1.1} has at most one
weak solution $(u,v) \in [W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)]^2, \frac{2N}{N+1}<p<N$.
\end{theorem}

\section{The proof of Theorem \ref{thm1.1}}

\begin{proof}   From (H2) and (H3), we know that
$[\alpha p,p^*)$ is not empty,
where $p^*=\frac{Np}{N-p}$.
Fix $q_0 \in [\alpha p,p^*)$, let $T:W^{1,q_0^{}}(\Omega)\mapsto
W^{1,q_0^{}}(\Omega) \cap L^\infty(\Omega)$ be the cut-off
function about ${\underline u,\overline u},{\underline v,\overline
v}$; i.e.,
\begin{gather*}
Tu(x)=\overline u(x), \quad  \overline u \le u, \\
              Tu(x)=u(x), \quad  \underline u \le u \le \overline u, \\
              Tu(x)=\underline u(x), \quad  u \le \underline u,\\
 Tv(x)=\overline v(x), \quad \overline v \le v, \\
              Tv(x)=v(x), \quad \underline v \le v \le \overline v, \\
              Tv(x)=\underline v(x), \quad v \le \underline v.
\end{gather*}
Next, we prove that $Tu, Tv\in W^{1,q_0^{}}(\Omega) \cap
L^\infty(\Omega)$. Firstly, we notice that
\begin{gather*}
|Tu(x)| \le \max \{|\underline u |,|\overline u | \}
=:M, \quad\text{a.e. }x\in  \Omega,\\
|Tv(x)| \le \max \{|\underline v |,|\overline v | \}
=: m, \quad\text{a.e. } x\in \Omega
\end{gather*}
for every $u,v \in W^{1,q_0}(\Omega)$, then $Tu,Tv \in
L^\infty(\Omega)$. Since the embedding of $W^{2,p}(\Omega)$ into
$W^{1,q_0}(\Omega)$ is compact and $\overline u,\underline
u,\overline v,\underline v \in W^{1,q_0} (\Omega)$, then by
\cite[A.6]{k1}, we know $|u-v|\in W^{1,q_0}(\Omega)$. Also, from
\begin{gather*}
Tu(x) = \frac{u+\overline u+2\underline u-|u-\overline u |}{4}
 + \frac{| u+\overline u-2
\underline u-|u-\overline u ||}{4},\\
Tv(x) = \frac{v+\overline v+2\underline v-|v-\overline v |}{4}
 + \frac{| v+\overline v-2
\underline v-|v-\overline v ||}{4}
\end{gather*}
we know that
$Tu,Tv\in W^{1,q_0}(\Omega)$, hence $Tu,Tv \in W^{1,q_0}(\Omega)
\cap L^\infty(\Omega)$.

Let  $S:[0,1] \times
[W^{1,q_0}(\Omega)]^2 \mapsto [W^{1,q_0}(\Omega)]^2$ be defined as
$S(t,u,v) = (w_1,w_2)$, where $(w_1, w_2)$ is the solution of the
following boundary-value problem
\begin{gather} \label{e2.1}
\begin{gathered}  L_1 w_1 = tf(x,Tu,Tv,D(Tu),D(Tv)), x\in \Omega,  \\
                w_1 = 0 , x\in \partial \Omega,
\end{gathered}\\
 \label{e2.2}
\begin{gathered} L_2 w_2 = tg(x,Tu,Tv,D(Tu),D(Tv)),  x\in \Omega,  \\
              w_2 = 0 ,  x\in \partial \Omega.
\end{gathered}
\end{gather}
According to (H1)-(H4), for every $u, v\in W^{1,q_0}(\Omega)$, we
have $f,g \in L^p(\Omega)$. Then, based on \cite[Theorem 6.4]{c1},
\eqref{e2.1} and \eqref{e2.2} have a unique solution $(w_1,
w_2)\in [W^{2,p}(\Omega) \cap W^{1,p}_0(\Omega)]^2 $ which means
that $S$ is a well-defined operator. Obviously $S(0,u,v) = (0,0)$,
then by the Sobolev embedding theorem, $ W^{2,p}(\Omega)
\hookrightarrow W^{1,q_0}(\Omega)$, we know that $S$ is
continuous.

Next we prove that $(u,v) \in [W^{1,q_0}(\Omega)]^2$, and for a
certain $t\in[0,1]$, $S(t,u,v)=(u,v)$ and this $(u,v)$ satisfies
$$
\| u \|_{1,q_0}+\| v \|_{1,q_0} \le C.
$$
According to the Sobolev embedding theorem and (H1), we have
\begin{equation} \label{e2.3}
\begin{aligned}
\|u \|_{1,q_{0}}
&\le C \|u \|_{2,p} \\
&\le C(\|h_1(x,Tu,Tv)\|_{p}+ k_{1} \||D(Tu)|^{\alpha_1}\|_{p}
+k_{2}\||D(Tv)|^{\alpha_2}\|_{p})
\end{aligned}
\end{equation}
and
\begin{equation} \label{e2.4}
 \begin{aligned} \|v \|_{1,q_{0}}
&\le C \|v \|_{2,p} \\
&\le C(\|h_{2}(x,Tu,Tv)\|_{p}
+ k_{3} \||D(Tu)|^{\alpha_3}\|_{p}+k_{4}\||D(Tv)|^{\alpha_4}\|_{p}).
\end{aligned}
\end{equation}
Then from the definition of $Tu$ and $Tv$, and the condition
(H2) we know that
\begin{equation} \label{e2.5}
\|h_{i}(x,Tu,Tv)\|_{p} \le C , \quad i = 1,2.
\end{equation}
Where $C$ depends only on $\overline u,\underline u,\overline
v,\underline v $ and $p$. When $i=1, 3$, we have
\begin{equation} \label{e2.6}
\|D(Tu)^{\alpha_{i}}\|_{p} =
[\|D(Tu)\|_{\alpha_{i}p}]^{\alpha_{i}} =\begin{cases}
\| D \overline u \|_{\alpha_{i}p}]^{\alpha_{i}}, & u \ge \overline u \\
[\| Du \|_{\alpha_{i}p}]^{\alpha_{i}}, &  \underline u \le u \le \overline u \\
[\| D \underline u \|_{\alpha_{i}p}]^{\alpha_{i}}, &
u \le \underline u.
\end{cases}
\end{equation}
When $i=2,4$, we have
\begin{equation} \label{e2.7}
||D(Tv)^{\alpha_{i}}||_{p} =
[\|D(Tv)\|_{\alpha_{i}p}]^{\alpha_{i}} =\begin{cases}
\| D \overline v \|_{\alpha_{i}p}]^{\alpha_{i}}, & v \ge \overline v \\
 [\| Dv \|_{\alpha_{i}p}]^{\alpha_{i}}, & \underline v \le v \le \overline v \\
 [\| D \underline v \|_{\alpha_{i}p}]^{\alpha_{i}}, &
 v \le \underline v.
\end{cases}
\end{equation}
Then by \cite[Theorem 4.14]{a1} (Ehrling-Nirenberg-Gagliardo), we
obtain
\begin{equation} \label{e2.8}
\begin{gathered} \|Du\|_{\alpha_{i}p} \le k_{1}\epsilon \|u \|_{2,
\alpha_{i}p}+k_{2}(\epsilon)\|u\|_{\alpha_{i}p},\
\underline u \le u \le
\overline u,\\
     \|Dv\|_{\alpha_{i}p} \le k_{3}\epsilon \|v \|_{2,\alpha_{i}p}+
           k_{4}(\epsilon)\|v\|_{\alpha_{i}p},\  \underline v \le v \le \overline v.
\end{gathered}
\end{equation}
Since $\underline u \le u \le \overline u,\underline v \le v \le
\overline v$, $\alpha_i \le 1$, $i=1,2,3,4$, by $\alpha_ip\leq q_0$
and $\overline u,\underline u,\overline v,\underline v \in
W^{1,q_{0}}(\Omega)$, we obtain
\begin{equation} \label{e2.9}
\begin{gathered}
\|u\|_{\alpha_{i}p} \le C,\quad
\|v\|_{\alpha_{i}p} \le C, \quad
\|D \overline u\|_{\alpha_{i}p} \le C,\\
\|D \underline u\|_{\alpha_{i}p} \le C, \quad
\|D \overline v\|_{\alpha_{i}p} \le C,\quad
\|D \underline v\|_{\alpha_{i}p} \le C, \\
\|u \|_{2,\alpha_{i}p} \le C\|u\|_{2,p},\quad
\|v \|_{2,\alpha_{i}p} \le C \|v\|_{2,p},\quad
i=1,2,3,4.
\end{gathered}
\end{equation}
Without loss of generality, we assume that
$\|u \|_{2,p} \ge 1,\|v \|_{2,p} \ge 1$.
By \eqref{e2.5},  \eqref{e2.6}, \eqref{e2.7}, \eqref{e2.8},
\eqref{e2.9}, we know from \eqref{e2.3} and \eqref{e2.4} that
$$
\|u \|_{2,p}+\|v \|_{2,p} \le C_{1} \varepsilon (\|u
\|_{2,p}+\|v\|_{2,p})+\frac{C}{2}.$$
 Select $\varepsilon =\frac{1}{2C_{1}}$, we can write
$$
\|u \|_{2,p}+\|v \|_{2,p}\le C.
$$
Then according to \eqref{e2.3} and \eqref{e2.4},
$$
\|u \|_{1,q_{0}^{}}+\|v \|_{1,q_{0}} \le C.
$$
 From the Leray-Schauder fixed point theorem
\cite[Theorem 11.3]{g1}, there exists a solution $(u,v) \in
[W^{1,q_{0}}(\Omega)]^2$ satisfying $S(1,u,v) = (u,v)$; i. e.,
\begin{equation} \label{e2.10}
 \begin{gathered}
      L_1 u = f(x,Tu,Tv,D(Tu),D(Tv)),\quad x\in \Omega, \\
      L_2 v = g(x,Tu,Tv,D(Tu),D(Tv)) ,\quad x\in  \Omega, \\
      u = v = 0,\quad x\in \partial \Omega.
 \end{gathered}
\end{equation}
Then $(u,v)\in [W^{1,q_0}(\Omega)]^2$ implies  $f,g \in
L^p(\Omega)$ and $(u,v) \in [W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)]^2$.
\par

Next we prove that $(u,v)$ satisfies
$$
\underline u \le u \le \overline u,  \underline v \le v \le \overline v.
$$
 Firstly we prove $u \le \overline u$.
Let $w = u- \overline u$, then $w \in W^{2,p}(\Omega)$, define
$w^+(x) = \max\{0,w(x)\}$, then we need only to prove $w^+ = 0$.
Previously,
\begin{gather*}
L_1 u = f(x,Tu,Tv,D(Tu),D(Tv)),\\
L_1 \overline u \geq f(x,\overline u,Tv, D\overline u, D(Tv)).
\end{gather*}
We obtain the inequality
\begin{equation} \label{e2.11}
L_1 w \leq [f(x,Tu,Tv,D(Tu),D(Tv))-f(x,\overline u,Tv,D\overline
u,D(Tv))].
 \end{equation}
 Multiply this inequality  by $w^{+}$, and integrate on $\Omega$.
On the left-hand side, we have
\begin{align*}
\int_{\Omega}L_1\omega\cdot\omega^+ &=\int_{\Omega}\sum_{i,j=1}^N
a_{ij}^1(x)D_i\omega\cdot D_j\omega^+ -\int_{\partial\Omega}\sum
a_{ij}^1(x)D_i\omega\cdot
\omega^+ \\
&=\int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega\cdot
D_j\omega^+
\end{align*}
Then we can rewrite \eqref{e2.11} as
\begin{equation} \label{e2.12}
\begin{aligned}
\int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega\cdot D_j\omega^+
&\leq \int_{\Omega}[f(x,Tu,Tv,D(Tu),D(Tv))\\
&\quad -f(x,\overline u,Tv,D\overline u,D(Tv))]w^{+}dx.
 \end{aligned}
\end{equation}

Let  $A=\{x \in \Omega : u(x) \le \overline u(x)\}$ and
$B=\{x \in\Omega : u(x)> \overline u(x)\}$. Then
$\Omega = A \cup B $.
Obviously on $A$, $w^{+}=0$. In $B$, $Tu=\overline{u}$. Then the
righthand side of \eqref{e2.12} is zero. That is,
$$
\int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega\cdot
D_j\omega^+=0.
$$
On $A$, $w^{+}=0$; on $B$, $\omega=\omega^+$. We can write the
previous equation as
$$
\int_{\Omega^+}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega^+\cdot
D_j\omega^+=0.
$$
Then according to the definition of the uniform elliptic operator,
$$
\lambda{|D\omega^+|}^2\leq\int_{\Omega^+}\sum_{i,j=1}^N
a_{ij}^1(x) D_i\omega^+\cdot D_j \omega^+=0.
$$
Consequently, $w^+ = 0, x\in \Omega$. That is in $\Omega$, $u \le
\overline u$. Similarly, we can prove that $\underline u \le u$
and $\underline v \le v\le \overline v$. From the definition of
$T$, we know $Tu=u$ and $Tv=v$. Then by \eqref{e2.10}, we obtain
that $(u,v) \in[W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)]^2$ is the
solution of \eqref{e1.1}. The proof is completed.
\end{proof}

\subsection*{An example}
In this section, we illustrate Theorem \ref{thm1.1}.
\begin{equation} \label{e3.1}
\begin{gathered}
L_1 u=\lambda_1\phi_1(x)+\frac{2\lambda_1}{9}u+v+\lambda_1\phi_1|Du|^{
\frac{1}{2}} ,  \quad  x \in \Omega, \\
L_2 v=\frac{3}{4}\lambda_2^2\phi_2(x)+\frac{\lambda_2^2}{12}u
+\frac{\lambda_2}{4}v+
\frac{\sqrt{3}}{4}\lambda_2^{\frac{3}{2}}\phi_2(x)|Dv|^{\frac{1}{2}},
\quad  x\in \Omega ,\\
u=v=0 , \quad x \in \partial \Omega.
\end{gathered}
\end{equation}
Here $\Omega$ is a regular domain in ${\mathbb \mathbb{R}}^N (N>2)$
with smooth boundary $\partial \Omega$, and
$$
\phi_i(x)=\frac{\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+\sup_{\Omega}
|D\varphi_i|} \le 1.
$$
In addition, $\lambda_i>0,\varphi_i(x)>0$ are the first eigenvalue
and the corresponding eigenfunction of operator $L_i$ in $\Omega$
with zero-Dirichlet boundary value condition. Therefore,
 $$
L_i\phi_i(x)=\frac{L_i\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+
\sup_{\Omega}|D\varphi_i|}
=\frac{\lambda_i\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+\sup_{\Omega}
|D\varphi_i|}=\lambda_i\phi_i(x).
$$
When $2<p<N$, we can verify that problem \eqref{e3.1} satisfies condition
(H1)--(H4). Let
$$
\underline u=0;\quad
\underline v=0;\quad
\overline u=9\phi;\quad
\overline v=3\lambda_1\phi;\quad
\phi=\max(\phi_1,\phi_2).
$$
It is not difficult to verify that $(\underline u,\overline
u),(\underline v,\overline v)$, based on this definition, is a
pair of super-solution and sub-solution for problem\eqref{e3.1}.
Hence according to Theorem \ref{thm1.1}, problem \eqref{e3.1} has
at least one solution $(u,v)\in [W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)]^2$.


\section{The proof of Theorem \ref{thm1.2}}

\begin{proof}
 Assume $(u_1,v_1),(u_2,v_2)\in [W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)]^2$
are solutions for problem \eqref{e1.1}; therefore
\begin{gather*}
L_1 u_1 = f(x,u_1,v_1,Du_1,Dv_1),\quad x \in  \Omega, \\
      L_2 v_1 = g(x,u_1,v_1,Du_1,Dv_1),\quad x \in \Omega, \\
      u_1 = v_1 = 0 ,\quad x \in \partial \Omega
\end{gather*}
and
\begin{gather*}
 L_1 u_2 = f(x,u_2,v_2,Du_2,Dv_2),\quad x \in \Omega, \\
      L_2 v_2 = g(x,u_2,v_2,Du_2,Dv_2) ,\quad x \in \Omega ,\\
      u_2 = v_2 = 0 ,\quad x \in \partial \Omega.
\end{gather*}
Then
\begin{gather}
L_1 (u_1-u_2) =f(x,u_1,v_1,Du_1,Dv_1)-f(x,u_2,v_2,Du_2,Dv_2),
  \label{e4.1} \\
L_2 (v_1-v_2) = g(x,u_1,v_1,Du_1,Dv_1)-g(x,u_2,v_2,Du_2,Dv_2),
  \label{e4.2} \\
(u_1-u_2)\mid_{\partial \Omega} = (v_1-v_2)\mid_{\partial \Omega}
= 0. \label{e4.3}
\end{gather}
Multiply \eqref{e4.1} by $(u_1-u_2)$ and \eqref{e4.2} by
$(v_1-v_2)$, and then integrate them on $\Omega$ yield
\begin{gather*}
\int_{\Omega}(u_1-u_2)\cdot L_1{(u_1-u_2)}
=\int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i{(u_1-u_2)}\cdot
D_j{(u_1-u_2)},
\\
\int_{\Omega}(v_1-v_2)\cdot L_2{(v_1-v_2)}
=\int_{\Omega}\sum_{i,j=1}^N a_{ij}^2(x)D_i{(v_1-v_2)}\cdot
D_j{(v_1-v_2)}.
\end{gather*}
By the uniformly elliptic condition, we get
\begin{gather*}
\int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i{(u_1-u_2)}\cdot
D_j{(u_1-u_2)}
\geq \lambda {\|Du_1-Du_2\|}^2, \\
\int_{\Omega}\sum_{i,j=1}^N a_{ij}^2(x)D_i{(v_1-v_2)}\cdot
D_j{(v_1-v_2)} \geq \lambda {\|Dv_1-Dv_2\|}^2.
\end{gather*}
Using the Lipschitz condition on $f,g$, it yields
\begin{align*}
&\int_{\Omega} (f(x,u_1,v_1,Du_1,Dv_1)-f(x,u_2,v_2,Du_2,Dv_2))(u_1-u_2)dx\\
&\leq L\int_{\Omega}(|u_1-u_2|+|v_1-v_2|+|Du_1-Du_2|
+|Dv_1-Dv_2|)\cdot |u_1-u_2|dx\\
&\le
L\int_{\Omega}(3{|u_1-u_2|}^2+{|v_1-v_2|}^2+\frac{{|Du_1-Du_2|}^2
+{|Dv_1-Dv_2|}^2}{2})dx
\end{align*}
and
\begin{align*}
&\int_{\Omega} (g(x,u_1,v_1,Du_1,Dv_1)-g(x,u_2,v_2,Du_2,Dv_2))(v_1-v_2)dx\\
&\leq L\int_{\Omega}(|u_1-u_2|+|v_1-v_2|+|Du_1-Du_2|+|Dv_1-Dv_2|)
  \cdot |v_1-v_2|dx\\
&\le
L\int_{\Omega}({|u_1-u_2|}^2+3{|v_1-v_2|}^2+\frac{{|Du_1-Du_2|}^2
 +{|Dv_1-Dv_2|}^2}{2})dx.
\end{align*}
Furthermore,
\begin{align*}
&\lambda {\|Du_1-Du_2\|}^2\\
&\leq \int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)
 D_i{(u_1-u_2)}\cdot D_j{(u_1-u_2)}\\
&\leq L\int_{\Omega}(3{|u_1-u_2|}^2+{|v_1-v_2|}^2
 +\frac{{|Du_1-Du_2|}^2+{|Dv_1-Dv_2|}^2}{2})dx
\end{align*}
and
\begin{align*}
&\lambda {\|Dv_1-Dv_2\|}^2\\
&\leq \int_{\Omega}\sum_{i,j=1}^N a_{ij}^2(x)
 D_i{(v_1-v_2)}\cdot D_j{(v_1-v_2)}\\
&\leq L\int_{\Omega}({|u_1-u_2|}^2+3{|v_1-v_2|}^2
 +\frac{{|Du_1-Du_2|}^2+{|Dv_1-Dv_2|}^2}{2})dx.
\end{align*}
Summing  these two formulas yields
\begin{equation} \label{e4.4}
\begin{aligned}
&\lambda {\|Du_1-Du_2\|}^2+\lambda {\|Dv_1-Dv_2\|}^2 \\
&\leq L\int_{\Omega}({4|u_1-u_2|}^2+4{|v_1-v_2|}^2
 +{|Du_1-Du_2|}^2+{|Dv_1-Dv_2|}^2)dx.
\end{aligned}
\end{equation}
Using the Poincar\'e inequality,
$$
\|u\|_{L^2(\Omega)}^2 \le C\|Du\|_{L^2(\Omega)}^2, \quad
\|v\|_{L^2(\Omega)}^2 \le C\|Dv\|_{L^2(\Omega)}^2.
$$
According to this formula and \eqref{e4.4}, we have
$$
\int_{\Omega}[|D(u_1-u_2)|^2+|D(v_1-v_2)|^2]\,dx \le L
\frac{4C+1}{\lambda}
\int_{\Omega}[|D(u_1-u_2)|^2+|D(v_1-v_2)|^2]\,dx
$$
By condition $(H5)$, $L  \frac{4C+1}{\lambda}<1$, we get
$D(u_1-u_2)=0,D(v_1-v_2)=0$, $x \in \Omega$. Since $u_i=v_i=0$ on
 $\partial \Omega$ for $i=1,2,$  it follows that
 $u_1=u_2$ and  $v_1=v_2$, a.e. $x\in \Omega $.
This completes the proof.
\end{proof}

\begin{thebibliography}{0}

\bibitem{a1} R. A. Adams;
\emph{Sobolev Space}. Academic Press, New York,1975.

\bibitem{a2} C. O. Alves, D. C. de Morais Filho and
M. A. S. Souto.
\emph{On systems of elliptic equations involving subcritical or critical
Sobolev exponents}, Nonlinear Analysis. 42 (2000), 771-787.

\bibitem{c1} Y. Z. Chen and L. c. Wu;
\emph{Elliptic partial differential
equations of second order and elliptic systems}. Science Press,
Beijing,1997.

\bibitem{c2} F. Correa;
\emph{Positive Solutions of a asymptotic planar system
of elliptic boundary Value Problems}. Inter. J. Math. and
Math.Sci. 21 (1998), 549-554.

\bibitem{d1} R. Dalmasso;
\emph{Existence and Uniqueness of positive solutions
of semilinear elliptic systems}, Nonlinear Analysis. 39 (2000)
559-568.

\bibitem{d2} M. Delgado and A. Suarez;
\emph{Existence of Solutions for
elliptic systems with holder continuous nonlinearities},
Differential and Integral Equations.  13 (2000), April-June,
453-477.

\bibitem{g1} D. Gilbarg and N. S. Trudinger;
\emph{Elliptic partial differential Equations of second order}.
Springer-Verlag, Berlin, 1983.

\bibitem{k1} D. Kinderlehrer, G. Stampacchia;
\emph{An introduction to
Variational Inequalities and their applications}. Academic Press,
New York 1980.

\bibitem{s1} D. H. Sattinger;
\emph{Monotone Methods in nonlinear Elliptic and
Parabolic boundary Value Problems}, Indiana Univ. Math. J. 21
(1972), 979-1000.


\end{thebibliography}

\end{document}