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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 81, pp. 1--10.\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-2009/81\hfil Existence of weak solutions]
{Existence of weak solutions for nonlinear systems involving several 
p-Laplacian operators}

\author[S. A. Khafagy, H. M. Serag\hfil EJDE-2009/81\hfilneg]
{Salah A. Khafagy, Hassan M. Serag} % in alphabetical order

\address{Salah A. Khafagy \newline
Mathematics Department, Faculty of Science, 
Al-Azhar University, Nasr City
11884, Cairo, Egypt}
\email{el\_gharieb@hotmail.com}

\address{Hassan M. Serag \newline
Mathematics Department, Faculty of Science, 
Al-Azhar University, Nasr City
11884, Cairo, Egypt}
\email{serraghm@yahoo.com}

\thanks{Submitted December 9, 2008. Published July 10, 2009.}
\subjclass[2000]{74H20, 35J65}

\keywords{Existence of weak solution; nonlinear system, p-Laplacian}

\begin{abstract}
 In this article, we study nonlinear systems involving several  p-Laplacian
 operators with variable coefficients. We consider  the system  
 \[
 -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i  
 -\sum_{j\neq i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha_j}u_j+f_i(x),  
 \]
 where $\Delta _{p}$ denotes the $p$-Laplacian defined by  
 $\Delta_{p}u\equiv \mathop{\rm div} [|\nabla u|^{p-2}\nabla u]$  
 with $p>1$, $p\neq 2$; $\alpha _i\geq 0$; $f_i$ are  given functions; 
 and the coefficients $a_{ij}(x)$ ($1\leq i,j\leq n$)  are bounded 
 smooth positive functions. We prove the existence of  
 weak solutions defined on bounded and unbounded domains 
 using the theory of nonlinear monotone operators.
\end{abstract}

\maketitle


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

\section{Introduction}

The generalized formulation of many boundary-value problems for partial
differential equations leads to operator equations of the form 
\[
A(u)=f 
\]
on a Banach space $V$. For this operator equation, we have the so-called
weak formulation:

\begin{quote}
Find $u\in V$ such that $(A(u),v)=(f,v)$ for all $v\in V$.
\end{quote}

Then functional analysis has tools for proving existence of generalized
(weak) solutions for a relatively wide class of differential equations that
appear in mathematical physics and industry.

The existence of weak solutions for $2\times 2$ nonlinear systems involving
several $p$-Laplacian operators have been proved, using the method of sub
and super solutions in \cite{S2005}, and using the theory of nonlinear
monotone operators in \cite{S2006}.

Here, we use the theory of nonlinear monotone operators to prove the
existence of weak solutions for the following nonlinear systems involving
several $p$-Laplacian operators with variable coefficients defined on a
bounded domain $\Omega $ of $\mathbb{R}^{N}$ with boundary $\partial \Omega $, 
\begin{gather*}
\begin{aligned} -\Delta _{p_i}u_i &\equiv -\mathop{\rm div} [|\nabla
u_i|^{p_i-2}\nabla u_i]\\ &=a_{ii}(x)|u_i|^{p_i-2}u_i-\sum_{j\neq
i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j+f_i(x) \quad
\text{in }\Omega , \end{aligned} \\
u_{i}=0,\quad i=1,2,\dots ,n,\quad \text{on }\partial \Omega .
\end{gather*}
Then, we generalize our results to systems defined on the whole space 
$\mathbb{R}^{N}$.

This article is organized as follow: In section 2 we introduce some
technical results and definitions concerning the theory of nonlinear
monotone operators. We study the existence of weak solutions for $n\times n$
nonlinear systems defined on a bounded domain in section 3, and on unbounded
domains in section 4.

\section{Preliminary results}

First, we introduce some results concerning the theory of nonlinear monotone
operators \cite{F1994}.

Let $A:V\to V^{\prime }$ be an operator on a Banach space $V$. We say that
the operator $A$ is:\newline
\emph{Bounded} if it maps bounded sets into bounded; i.e., for each $r>0$
there exists $M>0$ ($M$ depending on $r$) such that 
\[
\| u\| \leq r\text{ implies } \| A(u)\| \leq M,\quad \forall u\in V; 
\]
\emph{coercive} if $\lim_{\| u\| \to \infty } \langle A(u),u\rangle / \| u\|
=\infty $; \newline
\emph{monotone} if $\langle A(u_{1})-A(u_{2}),u_{1}-u_{2}\rangle \geq 0$ for
all $u_{1},u_{2}\in V$; \newline
\emph{strictly monotone} if $\langle A(u_{1})-A(u_{2}),u_{1}-u_{2}\rangle >0$
for all $u_{1},u_{2}\in V$, $u_{1}\neq u_{2}$; \newline
\emph{continuous} if $u_{k}\to u$ implies $A(u_{k})\to A(u)$, for all $\
u_{k},u\in V$; \newline
\emph{strongly continuous} if $u_{k}\overset{w}{\to }u$ implies $A(u_{k})\to
A(u)$, for all $u_{k},u\in V$; \newline
\emph{continuous on finite-dimensional subspaces} if $A:V_{n}\to
V_{n}^{\prime }$ is continuous for each subspace $V_{n}$ of finite
dimension. \newline
\emph{demicontinuous} if $u_{k}\to u$ implies $A(u_{k}) \overset{w}{\to }A(u)
$, for all $u_{k},u\in V$; \newline
the operator $A$ is said to be satisfy the $M_{0}$-condition if  $u_{k} 
\overset{w}{\to }u$, $A(u_{k})\overset{w}{\to }f$, and $[\langle
A(u_{k}),u_{k}\rangle \to \langle f,u\rangle ]$\ imply $A(u)=f$.

\begin{remark} \label{rmk1} \rm
\begin{itemize}
\item[(i)] Strongly continuous operators are continuous, and
 they are continuous on finite dimensional subspaces.
\item[(ii)] Strongly continuous
operators are bounded and satisfy the $M_{0}$-condition.
\item[(iii)] Strictly monotone operators are monotone operators.
\item[(iv)] Monotone and continuous operators satisfy the
$M_{0}$-condition.
\end{itemize}
\end{remark}

\begin{theorem} \label{thm2}
Let $V$ be a separable reflexive Banach space and
$A:V\to V'$ an operator which is: coercive, bounded, continuous on
finite-dimensional subspaces and satisfying the $M_{0}-$condition.
Then the equation $A(u)=f$ admits a solution for each $f\in V'$.
\end{theorem}

Next, we introduce the Sobolev space $W^{1,p}(\Omega ),1<p<\infty $, defined
as the completion of $C^{\infty }(\Omega )$ with respect to the norm (see 
\cite{A1975}) 
\begin{equation}
\| u\| _{W^{1,p}}=\Big[ \int_{\Omega }|\nabla u|^{p}+|u|^{p} \Big] 
^{1/p}<\infty .  \label{21}
\end{equation}

Since we are studying a Dirichlet problem, we define the space 
$W_{0}^{1,p}(\Omega )$ as the closure of $C_{0}^{\infty }(\Omega )$ in 
$W^{1,p}(\Omega )$ with respect to the norm 
\begin{equation}
\| u\| _{W_{0}^{1,p}}=\Big[ \int_{\Omega }|\nabla u|^{p}\Big] ^{1/p}<\infty ,
\label{22}
\end{equation}
which is equivalent to the norm given by (\ref{21}). Both spaces 
$W^{1,p}(\Omega )$ and $W_{0}^{1,p}(\Omega )$ are well defined reflexive
Banach Spaces. The space $W_{0}^{1,p}(\Omega )$ is compactly imbedded in the
space $L^{p}(\Omega )$; i.e., 
\begin{equation}
W_{0}^{1,p}(\Omega )\hookrightarrow \hookrightarrow L^{p}(\Omega ),
\label{23}
\end{equation}
which implies 
\begin{equation}
\| u\| _{L^{p}(\Omega )}\leq c\ \| u\| _{W_{0}^{1,p}(\Omega )}, \quad 
\text{i.e., } \int_{\Omega }a(x)|u|^{p}\leq c^{\prime }\int_{\Omega }|\nabla u|^{p}
\label{24}
\end{equation}
for every $u\in W_{0}^{1,p}(\Omega )$, where $a(x)$ is a smooth bounded
positive function.

Now, we introduce some results \cite{A1987} concerning the eigenvalue
problem 
\begin{equation}
\begin{gathered} -\Delta _{p}u\equiv -\mathop{\rm div}[|\nabla
u|^{p-2}\nabla u] =\lambda a(x)|u|^{p-2}u \quad \text{in }\Omega , \\ u=0
\quad \text{on }\partial \Omega . \end{gathered}  \label{25}
\end{equation}

We will say that $\lambda \in \mathbb{R}$ is an eigenvalue of (\ref{25}) if
there exists $u\in W_{0}^{1,p}(\Omega )$, $u\neq 0$, such that 
\[
\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla \varphi =\lambda \int_{\Omega
}a(x)|u|^{p-2}u\varphi 
\]
hods for all $\varphi \in W_{0}^{1,p}(\Omega )$. Then $u$ is called an
eigenfunction corresponding to the eigenvalue $\lambda $.

\begin{lemma} \label{lem3}
The eigenvalue problem \eqref{25} admits a positive principal eigenvalue
$\lambda =\lambda _{a}(\Omega )>0$ which is associated with a positive
eigenfunction $u\geq 0$ a.e. in $\Omega $ normalized by
$\| u\|_{p}=1 $. Moreover, the first eigenvalue is characterized by
\begin{equation}
\lambda _{a}(\Omega )=\inf \big\{\int_{\Omega }|\nabla
u|^{p}:\int_{\Omega }a(x)|u|^{p}=1\big\}.    \label{26}
\end{equation}
\end{lemma}

Also, from the characterization of the first eigenvalue given by \eqref{26},
we have 
\begin{equation}
\lambda _{a}(\Omega )\int_{\Omega }a(x)|u|^{p}\leq \int_{\Omega }|\nabla
u|^{p}.  \label{27}
\end{equation}

\section{Nonlinear systems defined on bounded domains}

Let us consider the nonlinear system 
\begin{equation}
\begin{gathered} -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i-\sum_{j\neq
i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j+f_i(x) \quad
\text{in }\Omega , \\ u_i=0,\quad i=1,2,\dots ,n, \quad \text{on } \partial
\Omega , \end{gathered}  \label{31}
\end{equation}
where $a_{ii}(x)$ is a smooth bounded positive function, $\Omega $ is a
bounded domain of $\mathbb{R}^{N}$, and 
\begin{gather}
\alpha _i\geq 0,\quad f_i\in L^{p_i^{\ast }}(\Omega ),  \label{32} \\
\frac{1}{p_i}+\frac{1}{p_i^{\ast }}=1,\quad \frac{\alpha _i+1}{p_i}
=\frac{1}{2},\quad i=1,2,\dots ,n.  \label{33}
\end{gather}

\begin{theorem} \label{thm4}
For $(f_i)\in \prod_{i=1}^{n}L^{p_i^{\ast }}(\Omega )$, there
exists a weak solution $(u_i)$ in the space
$\prod_{i=1}^{n}W_{0}^{1,p_i}(\Omega )$
for the system \eqref{31}, if
\begin{equation}
\lambda _{a_{ii}}(\Omega )>1,\quad i=1,2,\dots ,n.
\label{34}
\end{equation}
\end{theorem}

\begin{proof}
We transform the weak formulation of (\ref{31}) to the  operator form 
$(A-B)U=F$, where, $A$, $B$ and $F$ are operators defined on 
$\prod_{i=1}^{n}W_{0}^{1,p_i}(\Omega )$ by 
\begin{gather}
(AU,\Phi )\equiv (A(u_{1},u_{2},\dots ,u_{n}),(\phi _{1},\phi _{2},\dots
,\phi _{n}))=\sum_{i=1}^{n}\int_{\Omega }|\nabla u_i|^{p_i-2}\nabla
u_i\nabla \phi _i,  \label{35} \\
\begin{aligned} (BU,\Phi )
&\equiv (B(u_{1},u_{2},\dots ,u_{n}),(\phi
_{1},\phi _{2},\dots ,\phi _{n}))\\ 
&=\sum_{i=1}^{n}[\int_{\Omega
}a_{ii}(x)|u_i|^{p_i-2}u_i\phi _i-\sum_{j\neq i}^{n}\int_{\Omega
}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j\phi _i], \end{aligned}
\label{36} \\
(F,\Phi )\equiv ((f_{1},f_{2},\dots ,f_{n}),(\phi _{1},\phi _{2},\dots ,\phi
_{n}))=\sum_{i=1}^{n}\int_{\Omega }f_i\phi _i.  \label{37}
\end{gather}
Now, consider the operator $J$ defined by 
\begin{equation}
(J(u),\phi )=\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla \phi .  \label{38}
\end{equation}
This operator is bounded: Since 
\[
| (J(u),\phi )| \leq \int_{\Omega }|\nabla u|^{p-1}| \nabla \phi | , 
\]
using H\"{o}lder's inequality, we obtain 
\[
| (J(u),\phi )| \leq \Big[ \int_{\Omega }|\nabla u|^{p}\Big] ^{\frac{p-1}{p}}%
\Big[ \int_{\Omega }|\nabla \phi |^{p} \Big] ^{1/p}=\| u\|
_{W_{0}^{1,p}(\Omega )}^{p-1}\| \phi \| _{W_{0}^{1,p}(\Omega )}. 
\]
Also, we can prove that $J$ is continuous, let us assume that $u_{k}\to u$
in $W_{0}^{1,p}(\Omega )$. Then $\|u_{k}-u\| _{W_{0}^{1,p}(\Omega )}\to 0$,
so that $\| \nabla u_{k}-\nabla u\| _{L^{p}(\Omega )}\to 0$. Applying
Dominated Convergence Theorem, we obtain 
\[
\| (|\nabla u_{k}|^{p-2}\nabla u_{k}-|\nabla u|^{p-2}\nabla u)\|
_{L^{p}(\Omega )}\to 0, 
\]
and hence 
\[
\| J(u_{k})-J(u)\| _{_{L^{p}(\Omega )}}\leq \| (|\nabla u_{k}|^{p-2}\nabla
u_{k}-|\nabla u|^{p-2}\nabla u)\| _{L^{p}(\Omega )}\to 0. 
\]
Finally, $J$ is strictly monotone: 
\begin{align*}
(J(u_{1})-J(u_{2}),u_{1}-u_{2}) &= \int_{\Omega }|\nabla u_{1}|^{p-2}\nabla
u_{1}\nabla u_{1}+\int_{\Omega }|\nabla u_{2}|^{p-2}\nabla u_{2}\nabla u_{2}
\\
&\quad-\int_{\Omega }|\nabla u_{1}|^{p-2}\nabla u_{1}\nabla
u_{2}-\int_{\Omega }|\nabla u_{2}|^{p-2}\nabla u_{2}\nabla u_{1};
\end{align*}
using H\"{o}lder's inequality, we obtain 
\begin{align*}
&(J(u_{1})-J(u_{2}),u_{1}-u_{2}) \\
&\geq \int_{\Omega }|\nabla u_{1}|^{p}+\int_{\Omega }|\nabla u_{2}|^{p}-
\Big[ \int_{ \Omega }|\nabla u_{1}|^{p}\Big] ^{\frac{p-1}{p}} 
\Big[ \int_{\Omega }|\nabla u_{2}|^{p}\Big] ^{\frac{1}{p}} \\
&\quad -\Big[ \int_{\Omega }|\nabla u_{2}|^{p}\Big] ^{\frac{p-1}{p}} 
\Big[ \int_{\Omega }|\nabla u_{1}|^{p}\Big] ^{1/p} \\
&= \| u_{1}\| _{W_{0}^{1,p}(\Omega )}^{p}+\| u_{2}\| _{W_{0}^{1,p}(\Omega
)}^{p}-\| u_{1}\| _{W_{0}^{1,p}(\Omega )}^{p-1}\|
u_{2}\|_{W_{0}^{1,p}(\Omega )} -\| u_{2}\| _{W_{0}^{1,p}(\Omega )}^{p-1}\|
u_{1}\| _{W_{0}^{1,p}(\Omega )},
\end{align*}
and hence, 
\begin{align*}
&(J(u_{1})-J(u_{2}),u_{1}-u_{2}) \\
&\geq (\| u_{1}\|_{W_{0}^{1,p}(\Omega )}^{p-1}-\| u_{2}\|
_{W_{0}^{1,p}(\Omega )}^{p-1})(\| u_{1}\| _{W_{0}^{1,p}(\Omega )}-\| u_{2}\|
_{W_{0}^{1,p}(\Omega)}) >0.
\end{align*}
Now, $AU$ can be written as the sum of $J_{1}(u_{1}),J_{2}(u_{2}),\dots
,J_{n}(u_{n})$ where 
\[
(J_i(u_i),\phi _i)=\int_{\Omega }|\nabla u_i|^{p_{_i}-2}\nabla u_i\nabla
\phi _i,\quad i=1,2,\dots ,n, 
\]
and as above, the operators $J_{1}$, $J_{2},\dots $ and $J_{n}$ are bounded,
continuous and strictly monotone; so their sum, the operator $A$, will be
the same.

For the operator $B$, 
\[
B\colon \prod_{i=1}^{n}W_{0}^{1,p_i}(\Omega )\to
\prod_{i=1}^{n}L^{p_i}(\Omega ), 
\]
we can prove that it is a strongly continuous operator. To prove that, let
us assume that $u_{ik}\overset{w}{\to }u_i$ in $W_{0}^{1,p_i}(\Omega )$, 
$i=1,2,\dots ,n$. Then, using (\ref{23}), $(u_{ik})\to (u_i)$ in 
$\prod_{i=1}^{n}L^{p_i}(\Omega)$. By the Dominated Convergence Theorem, 
\begin{gather*}
a_{ii}(x)| u_{ik}| ^{p_i-2}u_{ik}\to a_{ii}(x)| u_i| ^{p_i-2}u_i \quad 
\text{in }L^{p_i}(\Omega ), \\
-a_{ij}(x)| u_{ik}| ^{\alpha _i}| u_{jk}| ^{\alpha _j}u_{jk}\to
-a_{ij}(x)| u_{_i}| ^{\alpha _i}| u_j| ^{\alpha _j}u_j\quad \text{in }
L^{p_j}(\Omega ),
\end{gather*}
Since 
\begin{align*}
(BU_{k}-BU,W) &= (B(u_{1k},u_{2k},\dots ,u_{nk})-B(u_{1},u_{2},\dots
,u_{n}),(w_{1},w_{2},\dots ,w_{n})) \\
&= \sum_{i=1}^{n}\Big[\int_{\Omega }a_{ii}(x)(| u_{ik}| ^{p_i-2}u_{ik}-|
u_i| ^{p_i-2}u_i)w_i \\
&\quad-\sum_{j\neq i}^{n}\int_{\Omega }a_{ij}(x)(| u_{ik}| ^{\alpha _i}|
u_{jk}| ^{\alpha _j}u_{jk}-| u_{_i}| ^{\alpha _i}| u_j| ^{\alpha
_j}u_j)w_i\Big],
\end{align*}
it follows that 
\begin{align*}
\| BU_{k}-BU\| 
&\leq \sum_{i=1}^{n}\Big[\| a_{ii}(x)(| u_{ik}|
^{p_i-2}u_{ik}-| u_i| ^{p_i-2}u_i)\| _{L^{p_i}(\Omega )} \\
&\quad +\sum_{j\neq i}^{n}\| a_{ij}(x)(| u_{ik}| ^{\alpha _i}| u_{jk}|
^{\alpha _j+1}-| u_{_i}| ^{\alpha _i}| u_j| ^{\alpha _j+1})\Big)\|
_{L^{p_i}(\Omega )}]\to 0.
\end{align*}
This proves that $B$ is a strongly continuous operators. According to Remark 
\ref{rmk1}, the operator $A-B$ satisfies the $M_{0}$-condition. Now, to
apply Theorem \ref{thm2}, it remains to prove that $A-B$ is a coercive
operator 
\begin{align*}
&((A-B)U,U) \\
&= \sum_{i=1}^{n}\int_{\Omega }|\nabla u_i|^{p_i}-\sum_{i=1}^{n}
\Big[\int_{\Omega }a_{ii}(x)| u_i| ^{p_i}-\sum_{j\neq i}^{n}\int_{\Omega
}a_{ij}(x)| u_{_i}| ^{\alpha _i+1}| u_j| ^{\alpha _j+1}\Big] \\
&\geq \sum_{i=1}^{n}\int_{\Omega }|\nabla
u_i|^{p_i}-\sum_{i=1}^{n}\int_{\Omega }a_{ii}(x)| u_i| ^{p_i}.
\end{align*}
Using (\ref{27}), we obtain 
\begin{align*}
((A-B)U,U) &\geq \sum_{i=1}^{n}\int_{\Omega }|\nabla
u_i|^{p_i}-\sum_{i=1}^{n}\frac{1}{\lambda _{a_{ii}}(\Omega )} \int_{\Omega
}| \bigtriangledown u_i|^{p_i} \\
&=\sum_{i=1}^{n}(1-\frac{1}{\lambda _{a_{ii}}(\Omega )} )\int_{\Omega }|
\bigtriangledown u_i| ^{p_i},
\end{align*}
and hence, 
\[
((A-B)U,U)\geq k\sum_{i=1}^{n}\| u_i\| _{W_{0}^{1,p_i}(\Omega )}^{p_i}=k\|
(u_i)\| _{\prod_{i=1}^{n}W_{0}^{1,p_i}(\Omega )}. 
\]
So that 
\[
((A-B)U,U)\to \infty \quad \text{as }\| (u_i)\|
_{\prod_{i=1}^{n}W_{0}^{1,p_i}(\Omega )}\to \infty . 
\]
This proves the coercivity condition and so, the existence of a weak
solution for systems \eqref{31}.
\end{proof}

\section{Nonlinear systems defined on $\mathbb{R}^{N}$}

We consider the nonlinear system 
\begin{equation}
\begin{gathered} -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i-\sum_{j\neq
i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j+f_i(x), \quad
x\in \mathbb{R}^{N}, \\ \lim_{| x| \to \infty }u_i(x)=0,\quad i=1,2,\dots
,n,\quad x\in \mathbb{R}^{N}. \end{gathered}  \label{41}
\end{equation}
We assume that $1<p_i<N$, $i=1,2,\dots ,n$, and the coefficients $a_{ii}(x)$
and $a_{ij}(x)$ are smooth bounded positive functions such that 
\begin{equation}
0<a_{ii}(x)\in L^{\frac{N}{p_i}}(\mathbb{R}^{N})\cap L^{\infty } (\mathbb{R}
^{N}), \quad 0<a_{ij}(x)\in L^{\frac{N}{\alpha _i+\alpha _j+2}}(\mathbb{R}
^{N})\cap L^{\infty }(\mathbb{R}^{N}).  \label{42}
\end{equation}
To discuss this problem, we need the following results which are studied in 
\cite{F1997} and that we recall briefly.

Let us introduce the Sobolev reflexive Banach space 
\[
D^{1,p}(\mathbb{R}^{N})=\{u\in L^{\frac{Np}{N-p}}(\mathbb{R}^{N}): \nabla
u\in (L^{p}(\mathbb{R}^{N}))^{n}\}, 
\]
which is defined as the completion of $C_{0}^{\infty }(\mathbb{R}^{N})$ with
respect to the norm 
\begin{equation}
\| u\| _{D^{1,p}(\mathbb{R}^{N})}=\Big[ \int_{\mathbb{R}^{N}}|\nabla u|^{p}
\Big] ^{1/p}<\infty .  \label{43}
\end{equation}
Moreover $D^{1,p}(\mathbb{R}^{N})$ is embedded continuously in the space 
$L^{\frac{Np}{N-p}}(\mathbb{R}^{N})$; that is, 
$D^{1,p}(\mathbb{R} ^{N})\hookrightarrow L^{\frac{Np}{N-p}} (\mathbb{R}^{N})$,
 which implies 
\begin{equation}
\| u\| _{L^{\frac{Np}{N-p}}(\mathbb{R}^{N})}\leq k\ \| u\| _{D^{1,p}
(\mathbb{R}^{N})}.  \label{44}
\end{equation}

\begin{lemma} \label{lem5}
The eigenvalue problem
\begin{equation}
\begin{gathered}
-\Delta _{_{P}}u\equiv -\mathop{\rm div}[|\nabla u|^{p-2}\nabla u]
=\lambda a(x)|u|^{p-2}u
\quad \text{in }\mathbb{R}^{N}, \\
u(x)\to 0\quad \text{as }|x|\to \infty ,\quad u>0
\quad \text{in } \mathbb{R}^{N},
\end{gathered} \label{45}
\end{equation}
admits a positive principal eigenvalue $\lambda =\lambda _{a}(\Omega )$
which is associated with a positive eigenfunction
$u\in D^{1,p}(\mathbb{R}^{N})$.
Moreover, the principal eigenvalue $\lambda _{a}(\Omega )$ is
characterized by
\begin{equation}
\lambda _{a}(\Omega )\int_{\mathbb{R}^{N}}a(x)|u|^{p}
\leq \int_{\mathbb{R}^{N}}|\nabla u|^{p},\quad
\forall \text{ }u\in D^{1,p}(\mathbb{R}^{N})
  \label{46}
\end{equation}
where
\begin{equation}
0<a(x)\in L^{\frac{N}{p}}(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N}).  
\label{47}
\end{equation}
\end{lemma}

In this section, we assume that 
\begin{equation}
\begin{gathered} \alpha _i\geq 0,\quad f_i\in
L^{\frac{Np_i}{N(p_i-1)+p_i}}(\mathbb{R}^{N}),\quad \alpha _i+\alpha
_j+2<N,1<p_i<n \\ \frac{1}{p_i}+\frac{1}{p_i^{\ast }}=1, \quad
\frac{\alpha _i+1}{p_i}=\frac{1}{2},\quad i=1,2,\dots ,n. \end{gathered}
\label{488}
\end{equation}

\begin{theorem} \label{thm6}
For $(f_i)\in \prod_{i=1}^{n}L^{\frac{Np_i}{N(p_i-1)+p_i}
}(\mathbb{R}^{N})$, there exists a weak solution $(u_i)$ in
$\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$ for system \eqref{41}, if
\begin{equation}
\lambda _{a_{ii}}(\Omega )>1,\quad i=1,2,\dots ,n.  \label{49}
\end{equation}
\end{theorem}

\begin{proof}
As in section 3, we transform the weak formulation of the system \eqref{41}
to the operator form $(A-B)U=F$, where, $A$, $B$ and $F$ are operators
defined on $\prod _{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$ by 
\begin{gather}
\begin{aligned} (AU,\Phi )&\equiv (A(u_{1},u_{2},\dots ,u_{n}),(\phi
_{1},\phi _{2},\dots ,\phi _{n}))\\
&=\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i-2}\nabla u_i\nabla
\phi _i=\sum_{i=1}^{n}(J_i(u_i),\phi _i), \end{aligned}  \label{410} \\
\begin{aligned} (BU,\Phi )&\equiv (B(u_{1},u_{2},\dots ,u_{n}),(\phi
_{1},\phi _{2},\dots ,\phi _{n}))\\ &=\sum_{i=1}^{n}[\int_{\mathbb{R}
^{N}}a_{ii}(x)|u_i|^{p_i-2}u_i\phi _i-\sum_{j\neq
i}^{n}\int_{\mathbb{R}^{N}}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha
_j}u_j\phi _i], \ \end{aligned}  \label{411} \\
(F,\Phi )\equiv ((f_{1},f_{2},\dots ,f_{n}),(\phi _{1},\phi _{2},\dots ,\phi
_{n}))=\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}f_i\phi _i.  \label{412}
\end{gather}
First, we prove that $A,B$ and $F$ are bounded operators on
 $\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$.

For the operator $A$, by using (\ref{410}) and applying Holder inequality,
we have 
\begin{align*}
| (AU,\Phi )| &\leq \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla
u_i|^{p_i-1}|\nabla \phi _i| \\
&\leq \sum_{i=1}^{n}\Big[ \int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i}\Big] 
^{(p_i-1)/p_i} \Big[ \int_{\mathbb{R} ^{N}}|\nabla \phi _i|^{p_i}\Big] 
^{1/p_i} \\
&= \sum_{i=1}^{n}\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{p_i-1}\| \phi _i\|
_{D^{1,p_i}(\mathbb{R} ^{N})} \\
&=\Big(\sum_{i=1}^{n}\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{p_i-1}\Big)
\Big(\| (\phi _i)\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})}\Big).
\end{align*}
This proves the boundedness of the operator $A$.

For the operator $B$, we have 
\begin{align*}
| (BU,\Phi )| 
&\leq \sum_{i=1}^{n}[\int_{\mathbb{R}^{N}}
a_{ii}(x)|u_i|^{p_i-1}|\phi _i| +\sum_{j\neq i}^{n}\int_{\mathbb{R}
^{N}}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j+1}|\phi _i|] 
\\
&\leq \sum_{i=1}^{n} 
\bigg[\Big(\int_{\mathbb{R}^{N}}a_{ii}(x) ^{\frac{N}{p}} \Big)
 ^{\frac{p}{N}} 
\Big(\int_{\mathbb{R}^{N}} |u_i|^{\frac{Np_i}{N-p_i}}\Big)
 ^{\frac{(p_i-1)(N-p_i)}{Np_i}} 
\Big(\int_{\mathbb{R}^{N}}|\phi _i|
 ^{\frac{Np_i}{N-p_i}}\Big)^{\frac{N-p_i}{Np_i}} 
 \\
&\quad +\sum_{j\neq i}^{n}\Big[\int_{\mathbb{R}^{N}}(a_{ij}(x))
^{\frac{ N}{\alpha _i+\alpha _j+2}}\Big] ^{\frac{\alpha _i+\alpha _j+2}{N}\ } \Big[ %
\int_{\mathbb{R}^{N}}|u_i|^{\frac{Np_i}{N-p_i}}\Big] ^{ \frac{\alpha
_i(N-p_i)}{Np_i} } \\
&\quad\times \Big[ \int_{\mathbb{R}^{N}}|u_j|^{\frac{Np_j}{N-p_j}}
\Big] ^{\frac{(\alpha _j+1)(N-p_j)}{Np_j}\ }\Big[ \int_{\mathbb{R}
^{N}}|\phi _i|^{\frac{Np_i}{N-p_i}}\Big] ^{\frac{N-p_i}{Np_i}} \bigg] \\
&\leq \sum_{i=1}^{n}\Big[k_i\| u_i\| _{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1}\|
\phi _i\| _{D^{1,p_i}(\mathbb{R}^{N})} \\
&\quad +\sum_{j\neq i}^{n}l_i\| u_i\| _{D^{1,p_i}(\mathbb{R}^{N})}^{\alpha
_i}\| u_j\| _{D^{1,p_j}(\mathbb{R}^{N})}^{\alpha _j+1}\| \phi _i\|
_{D^{1,p_i}(\mathbb{R}^{N})}\Big] \\
&= \Big[ \sum_{i=1}^{n}\Big[k_i\| u_i\| _{D^{1,p_i}(\mathbb{R}
^{N})}^{p_i-1}+\sum_{j\neq i}^{n}l_i\| u_i\| _{D^{1,p_i}(\mathbb{R}
^{N})}^{\alpha _i}\| u_j\| _{D^{1,pj}(\mathbb{R}^{N})}^{\alpha _j+1}\Big]
\Big] \\
&\quad\times \| (\phi _i)\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R} ^{N})}
\end{align*} 
For the operator $F$, we have 
$(F,\Phi)=\sum_{i=1}^{N}\int_{\mathbb{R}^{n}}f_i\phi _i$ and so 
\begin{align*}
| (F,\Phi )| 
&= \big| \sum_{i=1}^{N}\int_{\mathbb{R}^{n}}f_i\phi _i\big| \\
& \leq \sum_{i=1}^{N}\Big[ \int_{\mathbb{R}^{n}}|f_i|
^{\frac{np_i}{n(p_i-1)+p_i}}\Big] ^{\frac{n(p_i-1)+p_i}{np_i}}
\Big[\int_{\mathbb{R}^{n}}|\phi _i|^{\frac{np_i}{n-p_i}}\Big] 
 ^{\frac{n-p_i}{np_i}} \\
&= \sum_{i=1}^{N}(\| f_i\| _{L^{\frac{np_i}{ n(p_i-1)+p_i}}(\mathbb{R}
^{n})})\| (\phi _i)\| _{\prod_{i=1}^{N}D^{1,p_i}(\mathbb{R}^{n})}.
\end{align*}
Now, as in section 3, the operator $A$ defined by 
($AU,\Phi )=\sum_{i=1}^{n}(J_i(u_i),\Phi )$ is continuous. Also it is 
strictly monotone
on $\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$, since 
\begin{align*}
&(J_i(u_{1})-J_i(u_{2}),u_{1}-u_{2}) \\
&\geq (\| u_{1}\| _{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1} -\|
u_{2}\|_{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1})(\| u_{1}\| _{D^{1,p_i}
(\mathbb{R}^{N})}-\| u_{2}\| _{D^{1,p_i}(\mathbb{R} ^{N})})>0.
\end{align*}

For the operator $B$, we can prove that it is a strongly continuous operator
by using Dominated Convergence theorem and continuous imbedding property for
the space $\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$ into 
$\prod_{i=1}^{n}L^{\frac{Np_i}{N-p_i}}(\mathbb{R}^{N})$. To prove that, 
let us assume that $u_{ik}\overset{w}{\to }u_i$ in $D^{1,p_i}(\mathbb{R}^{N})$, 
$i=1,2,\dots ,n$. Then $(u_{ik})\to (u_i)$ in 
$\prod_{i=1}^{n}L^{\frac{Np_i}{N-p_i}}(\mathbb{R}^{N})$. Now, the sequence 
$(u_{ik})$ is bounded in $D^{1,p_i}(\mathbb{R}^{N})$, $i=1,2,\dots ,n$, 
then it is containing a subsequence again denoted
by $(u_{ik})$ converges strongly to $u_i$ in $L^{\frac{Np_i}{N-p_i}}(B_{r_{0}})$,
 $i=1,2,\dots ,n$, for any  bounded ball $B_{r_{0}}=\{x\in 
\mathbb{R}^{N}:\| x\| \leq r_{0}\}$. Since 
$u_{ik},u_i\in L^{\frac{Np_i}{N-p_i}}(B_{r_{0}})$, Then using the Dominated 
Convergence Theorem, we have 
\begin{gather*}
\| a_{ii}(x)(| u_{ik}| ^{p_i-2}u_{ik}-| u_i| ^{p_i-2}u_i)\|
 _{^{ \frac{Np_i}{N(p_i-1)+p_i}}}\to 0, \\
\| a_{ij}(x)(| u_{ik}| ^{\alpha _i-1}| u_{jk}| ^{\alpha _j+1}u_{jk}-| u_i|
^{\alpha _i-1}| u_j| ^{\alpha _j+1}u_j)\| _{^{\frac{Np_i}{N(p_i-1)+p_i}
}}\to 0,
\end{gather*}
for $i=1,2,\dots ,n$. Since 
\begin{align*}
((BU_{k}-BU),W) &= (B(u_{1k},u_{2k},\dots ,u_{nk})-B(u_{1},u_{2},\dots
,u_{n}),(w_{1},w_{2},\dots ,w_{n})) \\
&= \sum_{i=1}^{n}\Big[\int_{\mathbb{R}^{N}}a_{ii}(x)(| u_{ik}|
^{p_i-2}u_{ik}-| u_i| ^{p_i-2}u_i)w_i \\
&\quad -\sum_{j\neq i}^{n}\int_{\mathbb{R}^{N}}a_{ij}(x)(| u_{ik}| ^{\alpha
_i}| u_{jk}| ^{\alpha _j}u_{jk}-| u_{_i}| ^{\alpha _i}| u_j| ^{\alpha
_j}u_j)w_i\Big],
\end{align*}
it follows that 
\begin{align*}
&\| BU_{k}-BU\|_{\prod_{i=1}^{n}D^{1,p_i}(B_{r_{0}})} \\
&\leq \sum_{i=1}^{n}\Big[\| a_{ii}(x)(| u_{ik}| ^{p_i-2}u_{ik}-| u_i|
^{p_i-2}u_i)\| _{ \frac{Np_i}{N(p_i-1)+p_i}} \\
&\quad +\sum_{j\neq i}^{n}\| a_{ij}(x)(| u_{ik}| ^{\alpha _i}| u_{jk}|
^{\alpha _j+1}-| u_{_i}| ^{\alpha _i}| u_j| ^{\alpha _j+1})\| 
_{\frac{Np_i}{ N(p_i-1)+p_i}}\Big]\to 0.
\end{align*}
As in \cite{S2006}, we can prove that, the norm 
\[
\| BU_{k}-BU\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})} 
\]
tends strongly to zero and then the operator $B$ is strongly continuous.
According to Remark \ref{rmk1}, the operator $A-B$ satisfies the $M_{0}$%
-condition. Now, to apply Theorem \ref{thm2}, it remains to prove that the
operator $A-B$ is a coercive operator, 
\begin{align*}
&((A-B)U,U) \\
&= \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i}-\sum_{i=1}^{n}\Big[%
\int_{\mathbb{R} ^{N}}a_{ii}(x)| u_i| ^{p_i}-\sum_{j\neq i}^{n}
\int_{\mathbb{R}^{N}}a_{ij}(x)| u_{_i}| ^{\alpha _i+1}| u_j| ^{\alpha _j+1}
\Big] \\
&\geq \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla
u_i|^{p_i}-\sum_{i=1}^{n}\int_{\mathbb{R} ^{N}}a_{ii}(x)| u_i| ^{p_i}.
\end{align*}
Using (\ref{46}), we obtain 
\begin{align*}
((A-B)U,U) &\geq \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i}
-\sum_{i=1}^{n}\frac{1}{\lambda _{a_{ii}}(\Omega )} \int_{\mathbb{R}^{N}}|
\bigtriangledown u_i|^{p_i} \\
&=\sum_{i=1}^{n}(1-\frac{1}{\lambda _{a_{ii}}(\Omega )} )\int_{\mathbb{R}
^{N}}| \bigtriangledown u_i| ^{p_i}.
\end{align*}
From (\ref{49}), we deduce 
\[
((A-B)U,U)\geq k\sum_{i=1}^{n}\| u_i\| _{D^{1,p_i}(\mathbb{R}
^{N})}^{p_i}=k\| (u_i)\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})}. 
\]
So that $((A-B)U,U)\to \infty$ as 
$\|(u_i)\| _{\prod_{i=1}^{n}D^{1,p_i}(
\mathbb{R}^{N})}\to \infty$. 
This proves the coercivity condition and so,
the existence of a weak solution for systems \eqref{41}.
\end{proof}

\subsection*{Acknowledgments}
The authors wish to thank the anonymous referees for their interesting
remarks.

\begin{thebibliography}{9}
\bibitem{A1975} Adams, R.; \emph{Sobolev Spaces}, Academic Press, New York,
1975.

\bibitem{A1987} Anane, A.,; \emph{Simplicite et isolation de la premiere
valeur propre du }$p-$\emph{Laplacien avec poids}, Comptes Rendus Acad. Sc.
Paris, 305, 725-728, 1987.

\bibitem{F1997} Fleckinger, J.; Manasevich, R.; Stavrakakies, N.;  De
Thelin, F.; \emph{Principal Eigenvalues for some Quasilinear Elliptic
Equations on }$\mathbb{R}^{N}$, Advances in Diff. Eqns., Vol. 2, No. 6,
981-1003, 1997.

\bibitem{F1994} Francu, J; \emph{Solvability of Operator Equations}, Survey
Directed to Differential Equations, Lecture Notes of IMAMM 94, Proc. of the
Seminar ``Industerial Mathematics and Mathematical Modelling'' , Rybnik,
Univ. West Bohemia in Pilsen, Faculty of Applied Scinces, Dept. of Math.,
July, 4-8, 1994.

\bibitem{S2005} Serag, H. and El-Zahrani, E.; \emph{Maximum Principle and
Existence of Positive Solution for Nonlinear Systems on }$\mathbb{R}^{N}$,
Electron. J. Diff. Eqns., Vol. 2005, No. 85, 1-12, 2005.

\bibitem{S2006} Serag, H. and El-Zahrani, E.;  \emph{Existence of Weak
Solution for Nonlinear Elliptic Systems on} $\mathbb{R}^{N}$, Electron. J.
Diff. Eqns., Vol. 2006, No. 69, 1-10,2006.
\end{thebibliography}

\end{document}