\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 85, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/85\hfil Maximum principle and existence of positive solutions] {Maximum principle and existence of positive solutions for nonlinear systems on $\mathbb{R}^N$} \author[H. M. Serag, E. El-Zahrani\hfil EJDE-2005/85\hfilneg] {Hassan M. Serag, Eada A. El-Zahrani} % in alphabetical order \address{Hassan M. Serag \hfill\break Mathematics Department, Faculty of Science \\ Al-Azhar University\\ Nasr City (11884), Cairo, Egypt} \email{serraghm@yahoo.com} \address{Eada A. El-Zahrani \hfill\break Mathematics Department, Faculty of Science for Girls\\ Dammam, P. O. Box 838, Pincode 31113, Saudi Arabia} \email{eada00@hotmail.com} \date{} \thanks{Submitted May 19, 2005. Published July 27, 2005.} \subjclass[2000]{35B45, 35J55, 35P65} \keywords{Maximum principle; nonlinear elliptic systems; $p$-Laplacian; \hfill\break\indent sub and super solutions} \begin{abstract} In this paper, we study the following non-linear system on $\mathbb{R}^N$ \begin{gather*} -\Delta_pu=a(x)|u|^{p-2}u+b(x)|u|^{\alpha}|v|^{\beta}v+f\quad x\in \mathbb{R}^N\\ -\Delta_qv=c(x)|u|^{\alpha}|v|^{\beta}u+d(x)|v|^{q-2}v+g \quad x\in \mathbb{R}^N\\ \lim_{|x|\to\infty}u(x)=\lim_{|x|\to\infty}v(x)=0,\quad u,v>0\quad \text{in }\mathbb{R}^N \end{gather*} where $\Delta_pu=\mathop{\rm div}|\nabla u|^{p-2}\nabla u)$ with $p>1$ and $p\neq 2$ is the p-Laplacian", $\alpha,\beta>0$, $p,q>1$, and $f,g$ are given functions. We obtain necessary and sufficient conditions for having a maximum principle; then we use an approximation method to prove the existence of positive solution for this system. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} The operator $-\Delta_p$ occurs in problems arising in pure mathematics, such as the theory of quasiregular and quasiconformal mappings and in a variety of applications, such as non-Newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc (see \cite{d8,f2}). We are concerned with existence of positive solutions and with the following form of the maximum principle: If $f, g\ge 0$ then $u, v\ge 0$ for any solution $(u, v)$ of \eqref{S}. The maximum principle for linear elliptic systems with constant coefficients and the same differential operator in all the equations, have been studied in \cite{d1,d3}. Systems defined on unbounded domains and involving Schr\"odinger operators have been considered in \cite{a1,a2,s1}. In \cite{f4,f5}, the authors presented necessary and sufficient conditions for having the maximum principle and for existence of positive solutions for linear systems involving Laplace operator with variable coefficients. These results have been extended in \cite{f1}, to the nonlinear system $$\begin{gathered} -\Delta_pu_i=\sum^{n}_{j=1}a_{ij}|u_j|^{p-2}u_j+f_i\quad u_i\text{ in }\Omega\\ u_i=0\quad \text{on }\partial\Omega \end{gathered} \label{P}$$ In \cite{b3}, it has been proved the validity of the maximum principle and the existence of positive solutions for the following system defined on bounded domain $\Omega$ of $\mathbb{R}^N$, and with cooperative constant coefficients $a, b, c, d$: $$\label{Z} \begin{gathered} -\Delta_pu=a|u|^{p-2}u+b|u|^\alpha|v|^\beta v+f,\quad \text{in } \Omega\\ -\Delta_qv=c|u|^\alpha|v|^\beta u+d|v|^{p-2}v+g,\quad \text{in } \Omega\\ u=v=0\quad \text{on }\partial \Omega \end{gathered}$$ Here, we study system $$\label{S} \begin{gathered} -\Delta_pu=a(x)|u|^{p-2}u+b(x)|u|^{\alpha}|v|^{\beta}v+f\quad x\in \mathbb{R}^N\\ -\Delta_qv=c(x)|u|^{\alpha}|v|^{\beta}u+d(x)|v|^{q-2}v+g \quad x\in \mathbb{R}^N\\ \lim_{|x|\to\infty}u(x)=\lim_{|x|\to\infty}v(x)=0,\quad u,v>0\quad \text{in }\mathbb{R}^N\\ \end{gathered}$$ where $\Delta_pu=\mathop{\rm div}|\nabla u|^{p-2}\nabla u)$ with $p>1$ and $p\neq 2$ is the p-Laplacian", $\alpha,\beta>0$, $p,q>1$, and $f,g$ are given functions. System \eqref{S} is a generalization for \eqref{Z} to the whole space $\mathbb{R}^N$ and the coefficients $a(x), b(x), c(x), d(x)$ are variables. We obtain necessary and sufficient conditions on the coefficients for having a maximum principle for system \eqref{S}. Then using the method of sup and super solutions, we prove the existence of positive solutions under some conditions on the functions $f$ and $g$. This article is organized as follows: In section 2, we give some assumptions on the coefficients $a(x),b(x), c(x), d(x)$ and on the functions $f,g$ to insure the existence of a solution of \eqref{S} in a suitable Sobolev space. We also introduce some technical results and some notation, which are established in \cite{a3,b1,f2}. Section 3 is devoted to the maximum principle of system \eqref{S}, while section 4 is devoted to the existence of positive solutions. \section{Technical results} In this section, we introduce some technical results concerning the eigenvalue problem (see \cite{f2}) $$\label{E} \begin{gathered} -\Delta_pu=\lambda g(x)\Psi_p(u)\quad \quad \text{in }\mathbb{R}^N\\ u(x)\to 0 \text{ as }|x|\to\infty,\quad u>0 \text{ in }\mathbb{R}^N \end{gathered}$$ where $\Psi_p(u)=|u|^{p-2}u$ and $g(x)$ satisfies $$\label{e2.1} g(x)\in L^{N/p}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N), \quad g(x)\ge 0\text{ almost everywhere in }\mathbb{R}^N$$ For $10$ such that for all $u\in D^{1,p}$, $$\label{e2.2} \|u\|_{L^{Np/(N-p)}}\le k \|u\|_{D^{1,p}}$$ Clearly $D^{1,p}(\mathbb{R}^N)$ is a reflexive Banach space, which is embedded continuously in $L^{Np/(N-p)}(\mathbb{R}^N)$ (see \cite{d5}). \begin{lemma} \label{lem2.1} \begin{itemize} \item[(i)] If $\{u_n\}$ is a sequence in $D^{1,p}$, with $u_n\to u$ weakly, then there is a subsequence, denoted again by $\{u_n\}$, such that $B(u_n)\to B(u)$ \item[(ii)] If $B'(u)=0$, then $B(u)=0$, where $B(u)=\int_{\mathbb{R}^N}g(u)|u|^p dx$. \end{itemize} \end{lemma} \begin{theorem} \label{thm2.2} Let $g$ satisfy \eqref{e2.1}. Then \eqref{E} admits a positive first eigenvalue $\lambda_{g}(p)$. Moreover, it is characterized by $$\label{e2.3} \lambda_{g}(p)\int_{\mathbb{R}^N}g(u)|u|^p\le\|u\|^{p}_{D^{1,p}}$$ \end{theorem} \begin{theorem} \label{thm2.3} Let $g$ satisfy \eqref{e2.1}. Then \begin{itemize} \item[(i)] the eigenfunction associated to $\lambda_{g}(p)$ is of constant sign; i.e., $\lambda_{g}(p)$ is a principal eigenvalue. \item[(ii)] $\lambda_{g}(p)$ is the only eigenvalue of \eqref{E} which admits positive eigenfunction. \end{itemize} \end{theorem} \section{Maximum Principle} We assume that $11,\quad \lambda_d(q)>1 $$\label{e3.4b} \big(\lambda_a(p)-1\big)^{(\alpha+1)/p} \big(\lambda_d(q)-1\big)^{(\beta+1)/q} -1 >0$$ Conversely, if the maximum principle holds, then \eqref{e3.4a} holds and $$\label{e3.4c} [(\lambda_a(p)-1)]^{(\alpha+1)/p} [\lambda_d(q)-1]^{(\beta+1)/q}>\Theta \Big(\frac{b(x)}{a(x)}\Big)^{(\alpha+1)/p} \Big(\frac{c(x)}{d(x)}\Big)^{(\beta+1)/q}\,,$$ where $$\Theta=\frac{\inf_{x} \big(\phi^p/\psi^q\big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}} {\sup_{x}\big(\phi^p/\psi^q\big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}}$$ and$\phi$(respectively$\psi$) is the positive eigenfunction associated to$\lambda_a(p)$(respectively$\lambda_d(q)$). \end{theorem} \begin{proof} The necessary condition: Assume that$\lambda_a(p)\le 1$, then the function$f:=a(x)(1-\lambda_a(p))\phi^{p-1}$and$g:=0$are nonnegative; nevertheless$(-\phi, 0)$satisfies \eqref{S}, which contradicts the maximum principle. Similarly, if$\lambda_d(q)\le 1$, then the functions$g:=d(x)(1-\lambda_d(q))\psi^{q-1}$and$f:=0$are nonnegative; nevertheless$(0,-\psi)$satisfies \eqref{S}, which contradicts the maximum principle. Now suppose that$\lambda_a(p)>1$,$\lambda_d(q)>1$and \eqref{e3.4c} does not hold; i.e., $$[(\lambda_a(p)-1)]^{(\alpha+1)/p} [\lambda_d(q)-1]^{(\beta+1)/q} \leq \Theta \Big(\frac{b(x)}{a(x)}\Big)^{(\alpha+1)/p} \Big(\frac{c(x)}{d(x)}\Big)^{(\beta+1)/q}\,,$$ Then, there exists$\xi>0$such that $$\Big(a(x)\frac{(\lambda_a(p)-1)}{b(x)}\Big)^{(\alpha+1)/p} \Big(\frac{\phi^p}{\psi^q}\Big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}} \le\xi\le \frac{\big(\frac{c(x)}{d(x)}\big)^{(\beta+1)/q}}{(\lambda_d(q)-1)^{(\beta+1)/q}} \Big(\frac{\phi^p}{\psi^q}\Big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}$$ Let$\xi=\big(\frac{D^q}{C^p}\big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}$with$C,D >0. Then \begin{align*} &\Big(a(x)\frac{(\lambda_a(p)-1)}{b(x)}\Big)^{(\alpha+1)/p} \Big(\frac{(C\phi)^p}{(D\psi)^q}\Big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}\\ &\le 1\le \frac{\big(\frac{c(x)}{d(x)}\big)^{(\beta+1)/q}}{(\lambda_d(q)-1)^{(\beta+1)/q}} \Big(\frac{(C\phi)^p}{(D\psi)^q}\Big)^{\frac{\alpha+1}{p}\frac{\beta+1}{q}} \end{align*} So \begin{gather} a(x)(\lambda_a(p)-1) ((C\phi)^p )^{(\beta+1)/q}\le b(x)(D\psi)^{\beta+1}\label{e3.5}\\ d(x)(\lambda_d(q)-1) ((D\psi)^q )^{(\alpha+1)/p} \le c(x)(C\phi)^{\alpha+1}\label{e3.6} \end{gather} From the two expression above, we have \begin{gather*} a(x)(\lambda_a(p)-1)((C\phi))^{p-1}\le b(x)(D\psi)^{\beta+1}(C\phi)^\alpha,\\ d(x)(\lambda_d(q)-1)((D\psi)^{q-1})\le c(x)(C\phi)^{\alpha+1}(D\psi)^\beta\,. \end{gather*} Hence \begin{gather*} f=-a(x)(\lambda_a(p)-1)((C\phi))^{p-1}+b(x)(D\psi)^{\beta+1}+(C\phi)^\alpha\ge0, \\ g=-d(x)(\lambda_d(q)-1)((D\psi))^{q-1}+c(x)(D\psi)^{\beta}+(C\phi)^{\alpha+1}\ge0 \end{gather*} Sincef$and$g$are nonnegative functions, and$(-C\phi,-D\psi)$is a solution of \eqref{S} and the maximum principle does not hold. \end{proof} Now, we show that the condition is sufficient. Assume that \eqref{e3.4a} and \eqref{e3.4b} hold. If$(u,v)$is a solution of \eqref{S}, then for$f,g \ge 0$, we obtain by multiplying the first equation of \eqref{S} by$\overline{u}:=\max(0,-u)$and integrating over$\mathbb{R}^N: \begin{align*} -\int_{\mathbb{R}^N}|\nabla \overline{u}|^p &=-\int_{\mathbb{R}^N}a(x)|\overline{u}|^p +\int_{\mathbb{R}^N}b(x)|\overline{u}|^{\alpha+1}|v^+|^{\beta}v^+\\ &\quad -\int_{\mathbb{R}^N}b(x)|\overline{u}|^{\alpha+1}|v^-|^\beta v^- +\int_{\mathbb{R}^N}f u^-\,. \end{align*} Then $$\int_{\mathbb{R}^N}|\nabla \overline{u}|^p\le \int_{\mathbb{R}^N}a(x)|\overline{u}|^p+ \int_{\mathbb{R}^N}b(x)|\overline{u}|^{\alpha+1}|v^-|^{\beta+1}\,.$$ From \eqref{e2.3}, we get $$(\lambda_a(p)-1)\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\le\int_{\mathbb{R}^N} b(x)|\overline{u}|^{\alpha+1}|v^-|^{\beta+1}$$ Applying Holder inequality and using \eqref{e3.2}, we find $$(\lambda_a(p)-1)\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\le \Big(\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\Big)^{(\alpha+1)/p} \Big(\int_{\mathbb{R}^N}d(x)|v^-|^q\Big)^{(\beta+1)/q}$$ Hence \begin{align*} &\Big[(\lambda_a(p)-1)\Big(\int_{\mathbb{R}^N}a(x)|\overline{u} |^p\Big)^{(\beta+1)/q}\\ &-\Big(\int_{\mathbb{R}^N}d(x)|v^-|^q\Big)^{(\beta+1)/q}\Big] \Big(\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\Big)^{(\alpha+1)/p} \leq 0 \end{align*} If\big(\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\big)^{(\alpha+1)/p} =0$, then$\overline{u}=0$. If not, we have $$(\lambda_a(p)-1)^{(\alpha+1)/p} \Big(\int_{\mathbb{R}^N}a(x)|\overline{u}|^p\Big) ^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}\le \Big(\int_{\mathbb{R}^N}d(x)|v^-|^q\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}} \label{e3.7}$$ Similarly $$(\lambda_d(q)-1)^{(\beta+1)/q}\Big(\int_{\mathbb{R}^N}d(x)|\overline{v}|^q\Big) ^{\frac{\alpha+1}{p}\frac{\beta+1}{q}}\le \Big(\int_{\mathbb{R}^N}a(x)|u^-|^p\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}} \label{e3.8}$$ From \eqref{e3.7} and \eqref{e3.8}, we obtain $$\Big((\lambda_a(p)-1)^{(\alpha+1)/p} (\lambda_d(q)-1)^{(\beta+1)/q}-1\Big) \Big(\int_{\mathbb{R}^N}d(x)|\overline{v}|^q\int_{\mathbb{R}^N}a(x)|u^-|^p\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}}\le 0$$ From \eqref{e3.4b}, we have$\overline{u}=\overline{v}=0$and hence$u\ge 0$,$v\ge 0$i.e. the maximum principle holds. \begin{corollary} \label{coro0} If$p=q$, then the maximum principle holds for system \eqref{S} if and only if conditions \eqref{e3.4a} and \eqref{e3.4b} are satisfied. \end{corollary} \section{Existence of positive solutions} By an approximation method used in \cite{b2}, we prove now that the system \eqref{S} has a positive solution in the space$D^{1,p}\times D^{1,q}$. For$\epsilon \in (0,1)$, we introduce the system $$\label{Sep} \begin{gathered} -\Delta_p u_\epsilon= a(x)\frac{(|u_\epsilon|^{p-2}u_\epsilon)}{(1+|\epsilon^{1/p}u_\epsilon|^{p-1})}+ b(x)\frac{|v_\epsilon|^\beta v_\epsilon}{(1+|\epsilon^{1/q}v_\epsilon|^{\beta+1})} \frac{|u_\epsilon|^\alpha }{(1+|\epsilon^{1/p}u_\epsilon|^{\alpha})}+f \text{ in }\mathbb{R}^N \\ -\Delta_q v_\epsilon= d(x)\frac{(|v_\epsilon|^{q-2}v_\epsilon)}{(1+|\epsilon^{1/q}v_\epsilon|^{q-1})}+ c(x)\frac{|v_\epsilon|^\beta}{(1+|\epsilon^{1/q}v_\epsilon|^{\beta})} \frac{|u_\epsilon|^\alpha u_\epsilon }{(1+|\epsilon^{1/p}u_\epsilon|^{\alpha+1})}+g \text{ in } \mathbb{R}^N \\ \lim_{|x|\to \infty }u_\epsilon=\lim_{|x|\to \infty }v_\epsilon=0\,,\quad u_\epsilon\,,\,v_\epsilon>0 \quad \text{in }\mathbb{R}^N \end{gathered}$$ Letting$(\xi,\eta)=(u_\epsilon, v_\epsilon)$then system above can be written as \begin{gather*} -\Delta_p\xi=h(\xi,\eta)+f\quad \text{in } \mathbb{R}^N\\ -\Delta_q\eta=k(\xi,\eta)+g\quad \text{in } \mathbb{R}^N\\ \xi, \eta\to 0 \text{ as } |x|\to \infty\quad \xi, \eta>0\text{ in } \mathbb{R}^N \end{gather*} where \begin{gather*} h(\xi,\eta)=a(x)\frac{|\xi|^{p-2}\xi}{(1+|\epsilon^{1/p}\xi|^{p-1})} +b(x)\frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/q}\eta|^{\beta+1})} \frac{|\xi|^\alpha}{(1+|\epsilon^{1/p}\xi|^\alpha)}, \\ k(\xi,\eta)=c(x)\frac{|\eta|^\beta}{(1+|\epsilon^{1/q}\eta|^\beta)} \frac{|\xi|^\alpha\xi}{(1+|\epsilon^{1/p}\xi|^{\alpha+1})}+ d(x)\frac{|\eta|^{q-2}\eta}{(1+|\epsilon^{1/q}\eta|^{q-1})}\,. \end{gather*} From \eqref{e3.1} and \eqref{e3.2}$h(\xi,\eta)$,$k(\xi,\eta)$are bounded functions; i.e., there exists$M>0$such that$|h(\xi,\eta)|\le M$, and$|k(\xi,\eta)|\le M$for all$\xi, \eta$. Then, as in \cite{b3}, we can prove the following lemma. \begin{lemma} \label{lem4.1} If$(\xi_k, \eta_k)\to (\xi,\eta)$, weakly in$L^{Np/(N-p)}(\mathbb{R}^N)\times L^{Nq/(N-q)}(\mathbb{R}^N)$, then $$\Big\|a(x)\Big(\frac{|\xi_k(x)|^{p-2}\xi_k(x)}{(1+|\epsilon^{1/p}\xi_k(x)|^{p-1})} -\frac{|\xi(x)|^{p-2}\xi(x)}{(1+|\epsilon^{1/p}\xi(x)|^{p-1})}\Big) \Big\|_{L^{Np/(N(p-1)+p)}(\mathbb{R}^N)}\to 0 \label{lm.1}$$ $$\label{lm.2} \Big\| b(x) \Big( \frac{|\eta_k|^\beta \eta_k}{(1+|\epsilon^{1/q}\eta_k|^{\beta+1})} \frac{|\xi_k|^\alpha}{(1+|\epsilon^{1/p}\xi_k|^\alpha)}- \frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/p}\eta|^{\beta+1})} \frac{|\xi|^\alpha}{(1+|\epsilon^{1/p}\xi|^{\alpha})} \Big) \Big\|$$ approaches$0$under the norm of${L^{Np/(N(p-1)+p)}(\mathbb{R}^N)}$, a.e. in$\mathbb{R}^N$as$k$approaches infinity in$L^{Nq/(N(q-1)+q)}(\mathbb{R}^N)$. $$\label{lm.3} \Big\| d(x) \Big(\frac{|\eta_k(x)|^{q-2}\eta_k(x)}{(1 +|\epsilon^{1/q}\eta_k(x)|^{q-1})} - \frac{|\eta(x)|^{q-2}\eta(x)}{(1+|\epsilon^{1/q}\eta(x)|^{q-1})}\Big) \Big\| \to 0$$ $$\label{lm.4} \Big\| c(x) \Big(\frac{|\xi_k|^\alpha\xi_k}{(1 +|\epsilon^{1/p}\xi_k|^{\alpha+1})} \frac{|\eta_k|^\beta}{(1+|\epsilon^{1/q}\eta_k|^\beta)} - \frac{|\xi|^\alpha\xi}{(1+|\epsilon^{1/p}\xi|^{\alpha+1})} \frac{|\eta|^\beta}{(1+|\epsilon^{1/q}\eta|^\beta)}\Big) \Big\|\to 0$$ a.e. in$\mathbb{R}^N$as$k\to\infty$in$L^{Nq/(N(q-1)+q)}(\mathbb{R}^N)$\end{lemma} \begin{lemma} \label{lem4.2} System \eqref{Sep} has a solution$U_{\epsilon}=:(u_\epsilon,v_\epsilon)$in$D^{1,p}(\mathbb{R}^N) \times D^{1,q}(\mathbb{R}^N)$. \end{lemma} \begin{proof} We complete the proof in four steps: \noindent Step 1.\; Construction of sub-super solutions of \eqref{Sep}: Let$\xi^0\in D^{1,p}$(respectively$\eta^0 \in D^{1,q}$be a solution of $$-\Delta_p\xi^0=M+f \quad (\text{resp. } -\Delta_q\eta^0=M+g) \label{e4.1}$$ and let$\xi_0\in D^{1,p}$(respectively$\eta_0 \in D^{1,q}$be a solution of $$-\Delta_p\xi_0=-M+f\quad (\text{resp. } -\Delta_q\eta_0=-M+g)\label{e4.2}$$ Then$(\eta^0, \xi^0)$is a super solution of \eqref{Sep} and$\eta_0,\xi_0$is a sub solution since \begin{gather*} -\Delta_p\xi^0-h(\xi^0,\eta)-f\ge -\Delta_p\xi^0-M-f=0\quad \forall \eta\in [\eta_0,\eta^0] \\ -\Delta_p\xi_0-h(\xi_0,\eta)-f\le -\Delta_p\xi_0-M-f=0\quad \forall \eta\in [\eta_0,\eta^0] \\ -\Delta_p\eta^0-h(\xi,\eta^0)-g\ge -\Delta_q\eta^0-(M+g)=0\quad \forall \xi\in [\xi_0,\xi^0] \\ -\Delta_p\eta_0-h(\xi,\eta_0)-g\le -\Delta_q\eta_0-(M+g)=0\quad \forall \xi\in [\xi_0,\xi^0] \end{gather*} Let us assume that$K=[\xi_0,\xi^0]\times [\eta_0\times \eta^0]$. \noindent Step 2.\; Definition of the operator$T$: We define the operator$T:(\xi,\eta)\to (w,z)$, where$(w,z)$is the solution of the system $$\begin{gathered} -\Delta_p w=h(\xi,\eta)+f\quad \text{in } \mathbb{R}^N\\ -\Delta_qz=k(\xi,\eta)+g\quad \text{in } \mathbb{R}^N\\ w=z\to 0 \quad \text{as } |x|\to \infty \end{gathered}\label{e4.3}$$ \noindent Sept 3.\; Construction of an invariant set under$T$. We have to prove that$T(k)\subset K$: From \eqref{e4.1} and \eqref{e4.3}, we get $$-\Delta_p w-\Delta_p\xi^0\le h(\xi,\eta) -M\label{e4.4}$$ Multiplying this equation by$(w-\xi^0)^+$and integrating over$\mathbb{R}^N$, we obtain $$\int_{\mathbb{R}^N}[\Psi_p(\nabla w)-\Psi_p(\nabla\xi^0)][\nabla(w-\xi^0)^+] \le\int_{\mathbb{R}^N}(h(\xi,\eta)-M)(w-\xi^0)^+\le 0.$$ By monotonicity of$p$-Laplacian, we have$(w-\xi^0)^+=0$and hence$w\le \xi^0$. Similarly$w\ge \xi_0$, so that step is complete. \noindent Step 4.\;$T$is completely continuous: We prove that$Tmaps weakly convergent sequence to strongly convergence ones. From \eqref{e4.3}, we get \begin{align*} &-\Delta_p w_k-\Delta_p w\\ &=a(x)\Big[\frac{|\xi^{p-2}_{k}|\xi_k} {(1+|\epsilon^{1/p}\xi_k|^{p-1})}-\frac{|\xi|^{p-2}\xi}{(1 +|\xi^{1/p}\xi|^{p-1})}\Big]\\ &\quad+ b(x)\Big[\frac{|\eta_k|^\beta\eta_k}{(1 +|\epsilon^{1/q}|\eta^{\beta+1}_{k})} \frac{|\xi_k|^\alpha}{(1+|\epsilon^{1/p}\xi_k|^\alpha)}- \frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/q}\eta|^{\beta+1})} \frac{|\xi_i|^\alpha}{(1+|\epsilon^{1/p}\xi_i|^\alpha)}\Big] \end{align*} Multiplying by(w_k-w)$and integrating over$\mathbb{R}^N, we obtain \begin{aligned} & \int [\Psi_p(\nabla w_k)-\Psi_p(\nabla w)][\nabla(w_k-w)]\\ &=\int a(x)\Big[\frac{|\xi^{p-2}_{k}|} {(1+|\epsilon^{1/p}\xi_k|^{p-1})} -\frac{|\xi|^{p-2}\xi}{(1+|\xi^{1/p}\xi|^{p-1})}\Big](w_k-w)\\ &\quad +\int b(x)\Big[\frac{|\eta_k|^\beta\eta_k}{(1 +|\epsilon^{1/q}|\eta^{\beta+1}_{k})} \frac{|\xi_k|^\alpha}{(1+|\epsilon^{1/p}\xi_k|^\alpha)}\\ &\quad - \frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/q}\eta|^{\beta+1})} \frac{|\xi_i|^\alpha}{(1+|\epsilon^{1/p}\xi_i|^\alpha)}\Big](w_k-w) \end{aligned}\label{e4.5} Using H\"older's inequality, we obtain \begin{align*} & \int [\Psi_p(\nabla w_k)-\Psi_p(\nabla w)][\nabla(w_k-w)]\\ &\le \Big\| a(x)\Big[\frac{|\xi^{p-2}_{k}|} {(1+|\epsilon^{1/p}\xi_k|^{p-1})}-\frac{|\xi|^{p-2}\xi}{(1 +|\xi^{1/p}\xi|^{p-1})}\Big] \Big\|_{L^{Np/(N(p-1)+p}(\mathbb{R}^N)}\\ &\quad \times \|(w_k-w)\|_{L^{Np/(N-p)}{(\mathbb{R}^N)}}\\ &\quad + \Big\|b(x)\Big[\frac{|\eta_k|^\beta\eta_k}{(1 +|\epsilon^{1/q}|\eta^{\beta+1}_{k})} \frac{|\xi_k|^\alpha}{(1+|\epsilon^{1/p}\xi_k|^\alpha)}\\ &\quad -\frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/q}\eta|^{\beta+1})} \frac{|\xi_i|^\alpha}{(1+|\epsilon^{1/p}\xi_i|^\alpha)}\Big]\Big\|_{L^{Np/(N(p-1) +p}(\mathbb{R}^N)} %&\quad \times \|(w_k-w)\|_{L^{Np/(N-p)}{(\mathbb{R}^N)}} \end{align*} It is well known \cite{s2}, that $$|\xi-\xi'|^p\le c\{[|\xi|^{p-2}\xi-|\xi'|^{p-2}\xi'](\xi -\xi')\}^{\alpha/2}\{|\xi|^p+|\xi'|^p\}^{1-(\alpha/2)} \quad \forall \xi,\xi'\in \mathbb{R}^N\label{e4.6}$$ where\alpha=p$if$1\le p \le 2$and$\alpha=2$if$p>2$. From \eqref{e4.6} and the continuous imbedding of$D^{1,p}(\mathbb{R}^N)$in$L^{Np/(N-p)}(\mathbb{R}^N), we get \begin{align*} & \int_{\mathbb{R}^N}|\nabla(w_k-w)|^p\\ &\le k \Big(\Big\| a(x)\Big[\frac{|\xi^{p-2}_{k}|} {(1+|\epsilon^{1/p}\xi_k|^{p-1})}-\frac{|\xi|^{p-2}\xi}{(1 +|\xi^{1/p}\xi|^{p-1})}\Big] \Big\|_{L^{Np/(N(p-1)+p}(\mathbb{R}^N)}\\ &\quad + \Big\|b(x)\Big[\frac{|\eta_k|^\beta\eta_k}{(1+|\epsilon^{1/q}| \eta^{\beta+1}_{k})} \frac{|\xi_k|^\alpha}{(1+|\epsilon^{1/p}\xi_k|^\alpha)}\\ &\quad -\frac{|\eta|^\beta\eta}{(1+|\epsilon^{1/q}\eta|^{\beta+1})} \frac{|\xi_i|^\alpha}{(1+|\epsilon^{1/p}\xi_i|^\alpha)}\Big] \Big\|_{L^{Np/(N(p-1)+p}(\mathbb{R}^N)} \Big)\|(w_k-w)\| \end{align*} Applying lemma \ref{lem4.1}, we obtain \|w_k-w\|_{D^{1,p}}^{p-1}\to 0$as$k\to +\infty$in$D^{1,p}(\mathbb{R}^N) $which implies $$(w_k)\to (w) \quad \text{strongly as } k\to +\infty \text{ in } D^{1,p}(\mathbb{R}^N).$$ Similarly we can prove that $$(z_k)\to (z)\quad \text{as } k\to +\infty \text{ in } D^{1,p},$$ So$T$is completely continuous operator. Since$k$is a convex, bounded, closed in:$D^{1,p}\times D^{1,q}$, we can apply Schauder's Fixed Point Theorem and obtain the existence of a fixed point for$T$, which gives the existence of solution$U_{\epsilon}=:(u_\epsilon,v_\epsilon)$of$S_{\epsilon}$. \end{proof} Now, we can prove the existence of solution for system \eqref{S}. \begin{theorem} \label{thm4.3} If \eqref{e2.1}--\eqref{e2.3} are satisfied, then for any$f\in L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N)$,$g\in L^{\frac{Nq}{N(q-1)+q}}(\mathbb{R}^N)$, system \eqref{S} has a nonnegative solution$U=(u,v)$in the space:$D^{1,p}\times D^{1,q}$. \end{theorem} \begin{proof} This proof is done in three steps. \noindent Step 1.\; First we prove that$U_\epsilon=:(\epsilon^{1/p}u_\epsilon,\epsilon^{1/q}v_\epsilon)$is bounded in$D^{1,p}\times D^{1,q}$. Multiplying the first equation of \eqref{Sep} by$(\epsilon u_\epsilon)$and integrating over$\mathbb{R}^N, we obtain \begin{align*} &\int|\nabla \epsilon^{1/p}u_\epsilon|^p \\ & \le \int a (x)| \epsilon^{1/p}u_\epsilon| +\int b(x) |\epsilon^{1/p} u_{\epsilon}|+\epsilon^{1/p'}\int |\epsilon^{1/p} u_{\epsilon}| |f|\\ &\le \|a(x)\|_{N/p}\|\epsilon^{1/p} u_{\epsilon}\|_{Np/(N-p)}+\|b(x)\|_{Np/(N(p-1)+p)}\|\epsilon^{1/p} u_{\epsilon}\|_{Np/(N-p)} \\ &\quad +\epsilon^{1/p'}\|\epsilon^{1/p} u_{\epsilon}\|_{Np/(N-p)}\|f\|_{Np(N(p-1)+p)}\\ &\le M\|\epsilon^{1/p}u_\epsilon\|_{Np/(N-p)}\\ &\le k M\|\epsilon^{1/p}u_\epsilon\|_{D^{1,p}} \end{align*} so\|\epsilon^{1/p}u_\epsilon\|^{p-1}_{D^{1,p}}\le k M$which implies$U_{\epsilon}=:(\epsilon^{1/p}u_\epsilon)$is bounded in$D^{1,p}$. Similarly for$(\epsilon^{1/q}v_\epsilon)$. \noindent Step 2.\;$U_{\epsilon}=:(\epsilon^{1/p}u_\epsilon,\epsilon^{1/q}v_\epsilon )$converges to (0,0) strongly in$D^{1/p}\times D^{1/q}$. From step 1,$U_{\epsilon}=:(\epsilon^{1/p}u_\epsilon, \epsilon^{1/q}v_\epsilon )$converges weakly to$(u^*,v^*)$in$D^{1,p}\times D^{1,q}$and strongly in$L^{\frac{Np}{N-p}}(\mathbb{R}^N)\times L^{\frac{Nq}{N-q}}(\mathbb{R}^N)$. Multiplying the first equation of \eqref{Sep}, by$(\epsilon^{1/p'})$, we get $$-\Delta_p(\epsilon^{1/p}u_\epsilon)= a(x)\frac{|\epsilon^{1/p}u_\epsilon|^{p-2}(\epsilon^{1/p}u_\epsilon)}{(1 +|\epsilon^{1/p}u_\epsilon|^{p-1})} +b(x)\frac{(\epsilon^{1/q}v_\epsilon)^\beta \epsilon^{1/q}v_\epsilon}{1+|\epsilon^{1/q}v_\epsilon|^{\beta+1}} \frac{|(\epsilon^{1/p}u_\epsilon)|^\alpha}{1 +|(\epsilon^{1/p}u_\epsilon)|^\alpha} +f\epsilon^{1/p'}$$ Again using Lemma \ref{lem4.1}, we have $$a(x) \frac{|\epsilon^{1/p}u_\epsilon|^{p-2}(\epsilon^{1/p}u_\epsilon)} {(1+|\epsilon^{1/p}u_\epsilon|^{p-1})}\to a(x) \frac{|u^*|^{p-2}(u^*)}{(1+|u^*|^{p-1})} \quad \text{strongly in }L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N)$$ and similarly $$b(x)\frac{(\epsilon^{1/q}v_\epsilon)^\beta \epsilon^{1/q}v_\epsilon}{1+|\epsilon^{1/q}v_\epsilon|^{\beta+1}} \frac{|(\epsilon^{1/p}u_\epsilon)|^\alpha}{1 +|(\epsilon^{1/p}u_\epsilon)|^\alpha}\to b(x) \frac{|v^*|^\beta v^*}{(1+|v^*|^{\beta+1})}\frac{|u^*|^\alpha}{(1+|u^*|^\alpha)}$$ strongly in$L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N)$. Using a classical result in \cite{l1}, we have $$-\Delta_p(\epsilon^{1/p}u_\epsilon)\to -\Delta_p(u_*)\quad \text {strongly in }L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N).$$ So $$-\Delta_p(u^*)=a(x)\frac{|u^*|^{p-2}(u^*)}{(1+|u^*|^{p-1})}+ b(x)\frac{|v^*|^\beta v^*}{(1+|v^*|^{\beta+1})}\frac{|u^*|^\alpha}{(1+|u^*|^\alpha)} \label{e4.7}$$ Multiplying this equality by$(u^*)^-$and integrating over$\mathbb{R}^N$, then applying \eqref{e2.3} we get $$(\lambda_a(p)-1)\int a(x)|u^{*-}|^p\le \int |\nabla u^{*-}|^p\le \int d(x) |v^{*-}|^{\beta+1}|u^{*-}|^{\alpha+1}$$ Using H\"older inequality and \eqref{e2.2}, as in the proof of Theorem \ref{thm3.1}, we deduce $$(\lambda_a(p)-1)^{(\alpha+1)/p} \Big(\int a(x)|u^{*-}|^p\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}} \le \Big(\int d(x) |v^{*-}|^q\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}} \label{e4.8}$$ Similarly, from the second equation of \eqref{Sep}, we have $$(\lambda_d(q)-1)^{(\beta+1)/q}\Big(\int d(x)|v^{*-}|^q\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}} \le \Big(\int a(x) |u^{*-}|^p\Big)^{\frac{\beta+1}{q} \frac{\alpha+1}{p}}\label{e4.9}$$ Multiplying \eqref{e4.8} by \eqref{e4.9}, we obtain $$\Big((\lambda_d(q)-1)^{(\beta+1)/q} (\lambda_a(p)-1)^{(\alpha+1)/p} -1\Big)\Big(\int d(x) |v^{*-}|^q \int a(x) |u^{*-}|^p\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}}\le 0$$ From conditions \eqref{e3.4a} and \eqref{e3.4b}, we have$u^*_-=v^*_-=0$, which implies that$u^*, v^*\ge 0$. We show that$(u^*=v^*=0)$. Multiplying \eqref{e4.7} by$u^*$, we get $$\Big((\lambda_d(q)-1)^{(\beta+1)/q} (\lambda_a(p)-1)^{(\alpha+1)/p} -1\Big)\Big(\int d(x) |v^{*}|^q \int a(x) |u^{*}|^p\Big)^{\frac{\beta+1}{q}\frac{\alpha+1}{p}}\le 0$$ which implies that$u^*=v^*=0$, and step 2 is complete. \noindent Step 3.\;$(u_\epsilon,v_\epsilon)$is bounded in$D^{1,p}\times D^{1,q}$: Assume that$\|u_{\epsilon}\|_{D^{1,p}} \to \infty$or$\|v_{\epsilon}\|_{D^{1,q}} \to \infty$Set$t_{\epsilon}=\max(\|u_{\epsilon}\|_{D^{1,p}},\|v_{\epsilon}\|_{D^{1,q}} )$and$z_{\epsilon}=u_\epsilon t^{-1/p}_{\epsilon}$,$w_{\epsilon}=v_\epsilon t^{-1/q}_{\epsilon}$. Dividing the first equation of \eqref{Sep} by$(t^{1/p'}_{\epsilon})$and the second by$(t^{1/q'}_{\epsilon}), we have \begin{align*} -\Delta_p(z_\epsilon)=a(x)\frac{|z_\epsilon|^{p-2}z_\epsilon} {(1+|\epsilon^{1/p}u_\epsilon|^{p-1})}+b(x)\frac{|w_\epsilon\|^\beta w_\epsilon }{(1+|\epsilon^{1/q}v_\epsilon|)^{\beta+1}} \frac{|z_\epsilon|^\alpha}{(1+|\epsilon^{1/p}u_\epsilon|)^\alpha}+f t^{-1/p'}_{\epsilon}\\ -\Delta_q(w_\epsilon)=d(x)\frac{|w_\epsilon|^{q-2}w_\epsilon} {(1+|\epsilon^{1/q}v_\epsilon|^{q-1})}+c(x)\frac{|z_\epsilon\|^\alpha z_\epsilon }{(1+|\epsilon^{1/p}u_\epsilon|)^{\beta+1}} \frac{|w_\epsilon|^\beta}{(1+|\epsilon^{1/q}v_\epsilon|)^\alpha}+g t^{-1/q'}_{\epsilon} \end{align*} As above, we can prove that(z_\epsilon, w_\epsilon) \to (z,w)$strongly in$D^{1,p}\times D^{1,q}; and taking the limit, we obtain \begin{align*} -\Delta_p z=a(x)\Psi_p(z)+b(x)|w|^\beta|z|^\alpha w\\ -\Delta_q w=d(x)\Psi_q(w)+c(x)|w|^\beta|z|^\alpha z \end{align*} and we deducew=z=0$. Since there exists a sequence$(\epsilon_n)_{n\in N}$such that either$\|z_{\epsilon_n}\|=1$or$\|w_{\epsilon_n}\|=1$we obtain a contradiction. Hence$(u_\epsilon, v_\epsilon)$is bounded in$D^{1,p}\times D^{1,q}$, we can extract a subsequence denoted$(u_\epsilon, v_\epsilon)$which converges to$(u_0, v_0)$weakly in$D^{1,p}\times D^{1,q}$as$\epsilon \to 0$. By using similar procedure as above, we can prove that that$(u_\epsilon, v_\epsilon)$converges strongly to$(u_0, v_0)$in$D^{1,p}\times D^{1,q}$. Indeed, since$(\epsilon^{1/p}u_\epsilon(x), \epsilon^{1/q}v_\epsilon(x))\to (0,0)$a.e. on$\mathbb{R}^N$, then, as in \cite{b3}, we have \begin{gather*} a(x)\frac{|u_\epsilon(x)|^{p-2}u_\epsilon(x)}{(1+|\epsilon^{1/p}u_\epsilon(x)|^{p-1})}\to a(x) |u_0(x)|^{p-2}u_0(x) \quad \text{a.e. in }\mathbb{R}^N\text { as } \epsilon \to 0 , \\ a(x)\frac{|u_\epsilon|^{p-2}u_\epsilon}{(1+|\epsilon^{1/p}u_\epsilon|^{p-1})} \le M|u_\epsilon|^{p-1} \le M h^{p-1}\in L^{p^*}(\mathbb{R}^N), \\ b(x)\frac{|v_\epsilon|^\beta v_\epsilon}{(1+|\epsilon^{1/q}v_\epsilon|)^{\beta+1}} \frac{|u_\epsilon|^\alpha}{(1+|\epsilon^{1/p}u_\epsilon|)^{\alpha_i}}\to b(x) |v_0|^\beta v_0|u_0|^\alpha \quad \text{a.e. in } \mathbb{R}^N\text{ as }\epsilon \to 0 , \\ b(x)\frac{|v_\epsilon|^\beta v_\epsilon}{(1+|\epsilon^{1/q}v_\epsilon|)^{\beta+1}} \frac{|u_\epsilon|^\alpha}{(1+|\epsilon^{1/p}u_\epsilon|)^{\alpha}} \le M_2 h^\alpha l^{\beta+1} \in L^{p^*}(\mathbb{R}^N) \end{gather*} Hence from the Dominated Convergence Theorem and Lemma \ref{lem4.1}, we obtain \begin{gather*} \Big[\int_{\mathbb{R}^N} a(x) \Big(\frac{|u_\epsilon|^{p-2}u_\epsilon}{ (1+|\epsilon^{1/p}u_\epsilon|^{p-1})}\Big) -(|u_0|^{p-2}u_0)^{p^*}\Big]^{1/p^*}\to 0, \\ \Big[\int_{\mathbb{R}^N} d(x) \Big(\frac{|v_\epsilon|^{q-2}u_\epsilon}{(1 +|\epsilon^{1/q}v_\epsilon|^{p-1})}\Big) -(|v_0|^{p-2}v_0)^{q^*}\Big]^{1/q^*}\to 0, \\ \Big(\int_{\mathbb{R}^N}\Big(b(x)\frac{|v_\epsilon|^\beta v_\epsilon}{(1+|\epsilon^{1/q}v_\epsilon|)^{\beta+1}} \frac{|u_\epsilon|^\alpha}{(1+|\epsilon^{1/p}u_\epsilon|Big)^{\alpha}} -|v_0|^\beta v_0|u_0|^\alpha\Big)^{p^*}\Big)^{1/p^*} \to 0, \\ \Big(\int_{\mathbb{R}^N}\Big(c(x)\frac{|u_\epsilon|^\alpha u_\epsilon}{(1+|\epsilon^{1/p}u_\epsilon|)^{\alpha+1}} \frac{|v_\epsilon|^\beta}{(1+|\epsilon^{1/q}v_\epsilon|)^{\beta}}-|u_0|^\alpha u_0|v_0|^\beta\Big)^{q^*}\Big)^{1/q^*} \to 0, \end{gather*} as$\epsilon \to 0$. Therefore, passing to the limit,$(u_\epsilon, v_\epsilon)\to (u_0, v_0)we obtain from \eqref{Sep}, \begin{align*} -\Delta_p u_0=a(x)|u_0|^{p-2}u_0+b(x)|v_0|^\beta|u_0|^\alpha v_0+f\\ -\Delta_q v_0=d(x)|v_0|^{q-2}v_0+c(x)|u_0|^\alpha|v_0|^\beta u_0+g \end{align*} Hence(u_0,v_0)$satisfies \eqref{S}. \end{proof} We remark that if$\alpha=\beta=0$and$p=q=2$, we obtain the results presented in \cite{f4,f5}. \subsection*{Acknowledgement} The authors would like to express their gratitude to Professor J. Fleckinger (QREMAQ - Univ. Toulouse 1, France) for her constructive suggestions. \begin{thebibliography}{00} \bibitem{a1} Alziary, B. Cardoulis, I. and Fleckenger, J.; \emph{Maximum Principle and Existence of Solutions for Elliptic Systems Involving Schr\"odinger Operators}, Real Acad. Sc., Madrid, Vol. 91, No. 1 (1998), 47-52. \bibitem{a2} Alziary, B., Fleckinger, J. and Takac, P.; \emph{An Extension of Maximum and Anti-Maximum Principle to a Schr\"odinger Equation in$\mathbb{R}^N$} Vol. 5 (2001), 359-382. \bibitem{a3} Anane, A.; \emph{Simplicite et Isolution de la Preimer valeur Propre du p-Laplacian aves Poids}, Comptes Rendus Acad. Sc. Paris Vol. 305 (1987), 725-728. \bibitem{b1} Barles, G.; \emph{Remarks on Uniqueness Results of the First Eigenvalue of the p-Laplacian}, Ann. Fac. Sc. ToulouseVol. t. 9(1988), 65-75. \bibitem{b2} Boccardo, L. Fleckingeg, J. and De Thelin, F.; \emph{Existence of Solutions for Some Non-Linear Cooperative Systems and Some Applications}, Diff. and Int. Eqn.Vol. 7, No. 3(1994), 689-698. \bibitem{b3} Bouchekif, M., Serag, H. and De Thelin, F.; \emph{On Maximum Principle and Existence of Solutions for Some Nonlinear Elliptic Systems}, Rev. Mat. Apl. Vol. 16 (1995), 1-16. \bibitem{d1} De Figueredo, D. and Mitidieri, E.; \emph{A Maximum Principle for an Elliptic system and Applications to Semilinear Problem}, SIAM, J. Math. Anal Vol. 17(1986), 836-849. \bibitem{d2} De Figueiredo, D., and Mitidieri, E.; \emph{Maximum Principle for Linear Elliptic System}, Quaterno Matematico Dip. Sc. Mat., Univ. Trieste (1988) Vol. 177. \bibitem{d3} De Figueriedo, D., and Mitidieri, E.; \emph{Maximum Principle for Cooperative Elliptic System}, Comptes Rendus Acad. Sc. Paris, Vol. 310 (1990), 49-52. \bibitem{d4} Diaz, J. I. \emph{Nonlinear Partial Differential Equations and free Boundaries}, Pitman Publ. Program (1985). \bibitem{d5} Djellit, A. and Tas, S.; \emph{Existence of solutions for a class of Elliptic systems in$\mathbb{R}^N$involving p-Laplacian}, Electronic J. Diff. Eqs. Vol. 2003 (2003), No. 56, 1-8. \bibitem{d6} Do O, J. M. B.; \emph{Solutions to perturbed eigenvalue problems of the p-Laplacian in$\mathbb{R}^N$}, Electronic J. Diff. Eqns. Vol. 1997 (1997), No. 11, 1-15. \bibitem{d7} Drabek, P.; \emph{Nonlinear eigenvalue problem for p-Laplacian in$\mathbb{R}^N$}, Math. Nachr. Vol. 173 (1995), 131-139. \bibitem{d8} Drabek, P., Kufner, A. and Nicolosi, F.; \emph{Quasilinear elliptic equation with degenerations and singularities}, Walter de Gruyter, Bertin, New-York (1997). \bibitem{f1} Fleckinger, J., Hernandez J. and De Thelin, F.; \emph{On Maximum Principle and existence of Positive Solutions for Some Cooperative Elliptic Systems}, J. Diff. and Int. Eqns. Vol. 8 (1995), 69-85. \bibitem{f2} Fleckinger, J., Manasevich, R., Stavrakakies, N. and De Thelin, F.; \emph{Principle Eigenvalues for some Quasilinear Elliptic Equations on$\mathbb{R}^N$}, Advance in Diff. Eqns, Vol. 2, No. 6 (1997), 981-1003. \bibitem{f3} Fleckinger, J. Pardo R. and de Thelin F.; \emph{Four-Parameter bifurcation for a p-Lapalcian system}, Electronic J. Diff. Eqns. Vol. 2001 (2001) , No. 6, 1-15. \bibitem{f4} Fleckinger, J. and Serag, H.; \emph{On Maximum Princple and Existence of Solutions for Elliptic Systems on$\mathbb{R}^N$}, J. Egypt. Math. Soc. Vol. 2(1994), 45-51. \bibitem{f5} Fleckinger, J. and Serag, H.; \emph{Semilinear Cooperative Elliptic Systems on$\mathbb{R}^N$}, Rend. di Mat., Vol. Seri VII 15 Roma (1995), 89-108. \bibitem{h1} Huang, X. Y.; \emph{Eigenvalues of the p-Laplacian in$\mathbb{R}^N$with indefinite weight}, Comm. Math. Univ. Carolinae Vol. 36 (1995). \bibitem{l1} Lions, J. L.; \emph{Quelques Methodes de resolution des Problems aux Limit Nonlineare}, Dunod, Paris (1969). \bibitem{s1} Serag, H. M. and Qamlo, A. H.; \emph{On Elliptic Systems Involving Schr\"odinger Operators}, Mediterranean J. of Measurement and Control Vol. 1 , No. 2 (2005), 91-97. \bibitem{s2} Stavrakakis, N. M. and Zographopoulos, N. B.; \emph{Existence results for quasilinear elliptic systems in$\mathbb{R}^N\$}, Electronic J. Diff. Eqns. Vol. 1999 (1999), No. 39, 1-15. \bibitem{y1} Yu, L. S.; \emph{Nonlinear p-Laplacian Problems on Unbounded domains}, Proc. Amer. Math. Soc. Vol. 115 (1992), No. 4, 1037-1045. \bibitem{z1} Zeidler, E.; \emph{Nonlinear Functional Analysis and its Applications Vol. III }, Variational Methods and Optimization, Springer Verlag, Berlin (1990). \end{thebibliography} \end{document}