\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 $W^{1,p}(\Omega ),10$ 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 $$\lambda _{a}(\Omega )=\inf \big\{\int_{\Omega }|\nabla u|^{p}:\int_{\Omega }a(x)|u|^{p}=1\big\}. \label{26}$$ \end{lemma} Also, from the characterization of the first eigenvalue given by \eqref{26}, we have $$\lambda _{a}(\Omega )\int_{\Omega }a(x)|u|^{p}\leq \int_{\Omega }|\nabla u|^{p}. \label{27}$$ \section{Nonlinear systems defined on bounded domains} Let us consider the nonlinear system $$\begin{gathered} -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i-\sum_{j\neq i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j+f_i(x) \quad \text{in }\Omega , \\ u_i=0,\quad i=1,2,\dots ,n, \quad \text{on } \partial \Omega , \end{gathered} \label{31}$$ 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 $$\lambda _{a_{ii}}(\Omega )>1,\quad i=1,2,\dots ,n. \label{34}$$ \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 $$(J(u),\phi )=\int_{\Omega }|\nabla u|^{p-2}\nabla u\nabla \phi . \label{38}$$ 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 $$\begin{gathered} -\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i-\sum_{j\neq i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j+f_i(x), \quad x\in \mathbb{R}^{N}, \\ \lim_{| x| \to \infty }u_i(x)=0,\quad i=1,2,\dots ,n,\quad x\in \mathbb{R}^{N}. \end{gathered} \label{41}$$ We assume that $10 \quad \text{in } \mathbb{R}^{N}, \end{gathered} \label{45} admits a positive principal eigenvalue$\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 $$\lambda _{a}(\Omega )\int_{\mathbb{R}^{N}}a(x)|u|^{p} \leq \int_{\mathbb{R}^{N}}|\nabla u|^{p},\quad \forall \text{ }u\in D^{1,p}(\mathbb{R}^{N}) \label{46}$$ where 01,\quad i=1,2,\dots ,n. \label{49} \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$\prod _{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})by \begin{gather} \begin{aligned} (AU,\Phi )&\equiv (A(u_{1},u_{2},\dots ,u_{n}),(\phi _{1},\phi _{2},\dots ,\phi _{n}))\\ &=\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i-2}\nabla u_i\nabla \phi _i=\sum_{i=1}^{n}(J_i(u_i),\phi _i), \end{aligned} \label{410} \\ \begin{aligned} (BU,\Phi )&\equiv (B(u_{1},u_{2},\dots ,u_{n}),(\phi _{1},\phi _{2},\dots ,\phi _{n}))\\ &=\sum_{i=1}^{n}[\int_{\mathbb{R} ^{N}}a_{ii}(x)|u_i|^{p_i-2}u_i\phi _i-\sum_{j\neq i}^{n}\int_{\mathbb{R}^{N}}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j}u_j\phi _i], \ \end{aligned} \label{411} \\ (F,\Phi )\equiv ((f_{1},f_{2},\dots ,f_{n}),(\phi _{1},\phi _{2},\dots ,\phi _{n}))=\sum_{i=1}^{n}\int_{\mathbb{R}^{N}}f_i\phi _i. \label{412} \end{gather} First, we prove thatA,B$and$F$are bounded operators on$\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})$. For the operator$A, by using (\ref{410}) and applying Holder inequality, we have \begin{align*} | (AU,\Phi )| &\leq \sum_{i=1}^{n}\int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i-1}|\nabla \phi _i| \\ &\leq \sum_{i=1}^{n}\Big[ \int_{\mathbb{R}^{N}}|\nabla u_i|^{p_i}\Big] ^{(p_i-1)/p_i} \Big[ \int_{\mathbb{R} ^{N}}|\nabla \phi _i|^{p_i}\Big] ^{1/p_i} \\ &= \sum_{i=1}^{n}\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{p_i-1}\| \phi _i\| _{D^{1,p_i}(\mathbb{R} ^{N})} \\ &=\Big(\sum_{i=1}^{n}\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{p_i-1}\Big) \Big(\| (\phi _i)\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N})}\Big). \end{align*} This proves the boundedness of the operatorA$. For the operator$B, we have \begin{align*} | (BU,\Phi )| &\leq \sum_{i=1}^{n}[\int_{\mathbb{R}^{N}} a_{ii}(x)|u_i|^{p_i-1}|\phi _i| +\sum_{j\neq i}^{n}\int_{\mathbb{R} ^{N}}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha _j+1}|\phi _i|] \\ &\leq \sum_{i=1}^{n} \bigg[\Big(\int_{\mathbb{R}^{N}}a_{ii}(x) ^{\frac{N}{p}} \Big) ^{\frac{p}{N}} \Big(\int_{\mathbb{R}^{N}} |u_i|^{\frac{Np_i}{N-p_i}}\Big) ^{\frac{(p_i-1)(N-p_i)}{Np_i}} \Big(\int_{\mathbb{R}^{N}}|\phi _i| ^{\frac{Np_i}{N-p_i}}\Big)^{\frac{N-p_i}{Np_i}} \\ &\quad +\sum_{j\neq i}^{n}\Big[\int_{\mathbb{R}^{N}}(a_{ij}(x)) ^{\frac{ N}{\alpha _i+\alpha _j+2}}\Big] ^{\frac{\alpha _i+\alpha _j+2}{N}\ } \Big[ % \int_{\mathbb{R}^{N}}|u_i|^{\frac{Np_i}{N-p_i}}\Big] ^{ \frac{\alpha _i(N-p_i)}{Np_i} } \\ &\quad\times \Big[ \int_{\mathbb{R}^{N}}|u_j|^{\frac{Np_j}{N-p_j}} \Big] ^{\frac{(\alpha _j+1)(N-p_j)}{Np_j}\ }\Big[ \int_{\mathbb{R} ^{N}}|\phi _i|^{\frac{Np_i}{N-p_i}}\Big] ^{\frac{N-p_i}{Np_i}} \bigg] \\ &\leq \sum_{i=1}^{n}\Big[k_i\| u_i\| _{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1}\| \phi _i\| _{D^{1,p_i}(\mathbb{R}^{N})} \\ &\quad +\sum_{j\neq i}^{n}l_i\| u_i\| _{D^{1,p_i}(\mathbb{R}^{N})}^{\alpha _i}\| u_j\| _{D^{1,p_j}(\mathbb{R}^{N})}^{\alpha _j+1}\| \phi _i\| _{D^{1,p_i}(\mathbb{R}^{N})}\Big] \\ &= \Big[ \sum_{i=1}^{n}\Big[k_i\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{p_i-1}+\sum_{j\neq i}^{n}l_i\| u_i\| _{D^{1,p_i}(\mathbb{R} ^{N})}^{\alpha _i}\| u_j\| _{D^{1,pj}(\mathbb{R}^{N})}^{\alpha _j+1}\Big] \Big] \\ &\quad\times \| (\phi _i)\| _{\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R} ^{N})} \end{align*} For the operatorF$, we have$(F,\Phi)=\sum_{i=1}^{N}\int_{\mathbb{R}^{n}}f_i\phi _iand so \begin{align*} | (F,\Phi )| &= \big| \sum_{i=1}^{N}\int_{\mathbb{R}^{n}}f_i\phi _i\big| \\ & \leq \sum_{i=1}^{N}\Big[ \int_{\mathbb{R}^{n}}|f_i| ^{\frac{np_i}{n(p_i-1)+p_i}}\Big] ^{\frac{n(p_i-1)+p_i}{np_i}} \Big[\int_{\mathbb{R}^{n}}|\phi _i|^{\frac{np_i}{n-p_i}}\Big] ^{\frac{n-p_i}{np_i}} \\ &= \sum_{i=1}^{N}(\| f_i\| _{L^{\frac{np_i}{ n(p_i-1)+p_i}}(\mathbb{R} ^{n})})\| (\phi _i)\| _{\prod_{i=1}^{N}D^{1,p_i}(\mathbb{R}^{n})}. \end{align*} Now, as in section 3, the operatorA$defined by ($AU,\Phi )=\sum_{i=1}^{n}(J_i(u_i),\Phi )$is continuous. Also it is strictly monotone on$\prod_{i=1}^{n}D^{1,p_i}(\mathbb{R}^{N}), since \begin{align*} &(J_i(u_{1})-J_i(u_{2}),u_{1}-u_{2}) \\ &\geq (\| u_{1}\| _{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1} -\| u_{2}\|_{D^{1,p_i}(\mathbb{R}^{N})}^{p_i-1})(\| u_{1}\| _{D^{1,p_i} (\mathbb{R}^{N})}-\| u_{2}\| _{D^{1,p_i}(\mathbb{R} ^{N})})>0. \end{align*} For the operatorB$, 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 operatorB$is strongly continuous. According to Remark \ref{rmk1}, the operator$A-B$satisfies the$M_{0}$% -condition. Now, to apply Theorem \ref{thm2}, it remains to prove that the operator$A-Bis 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}