\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 69, 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 (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/69\hfil Existence of weak solutions] {Existence of weak solutions for nonlinear elliptic systems on $\mathbb{R}^N$} \author[E. A. El-Zahrani, H. M. Serag, \hfil EJDE-2006/69\hfilneg] {Eada A. El-Zahrani, Hassan M. Serag} % in alphabetical order \address{Eada A. El-Zahrani \newline Mathematics Department, Faculty of Science for Girls,\\ Dammam, P. O. Box 838, Pincode 31113, Saudi Arabia} \address{Hassan M. Serag \newline Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City (11884), Cairo, Egypt} \email{serraghm@yahoo.com} \date{} \thanks{Submitted February 9, 2006. Published July 6, 2006.} \subjclass[2000]{35B45, 35J55} \keywords{Weak solutions; nonlinear elliptic systems; p-Laplacian; \hfill\break\indent monotone operators} \begin{abstract} In this paper, we consider the nonlinear elliptic system \begin{gather*} -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f,\\ -\Delta_qv=-c(x)|u|^\alpha |v|^\beta u + d(x) |v|^{q-2}v +g ,\\ \lim_{|x|\to\infty}u=\lim_{|x|\to\infty}v=0\quad u,v>0 \end{gather*} on a bounded and unbounded domains of $\mathbb{R}^N$, where $\Delta_p$ denotes the p-Laplacian. The existence of weak solutions for these systems is proved using the theory of monotone operators \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \def\q{\quad} \def\p{\partial} \section{Introduction} The generalized (the so-called weak) formulation of many stationary boundary-value problems for partial differential equations leads to operator equation of type $$ A(u)=f $$ on a Banach space. Indeed, the weak formulation consists in looking for an unknown function $u$ from a Banach space $V$ such that an integral identity containing $u$ holds for each test function $v$ from the space $V$. Since the identity is linear in $v$, we can take its sides as values of continuous linear functionals at the element $v\in V$. Denoting the terms containing unknown $u$ as the value of an operator $A$, we obtain $$ (A(u),v)=(f,v)\quad \forall v\in V, $$ which is equivalent to equality of functionals on $V$, i.e. the equality of elements of $V'$ (the dual space of $V$): $A(u)=f$ . Functional analysis yields tools for proving existence of generalized (weak) solutions to a relatively wide class of differential equations that appear in mathematical physics and industry. In our work, we consider nonlinear systems with model $A$ of the form $$ A\{u,v\}=\{-\Delta_pu-a(x)|u|^{p-2}u+b(x)|u|^\alpha|v|^\beta v, -\Delta_qv+c(x)|u|^\alpha |v|^\beta u - d(x) |v|^{q-2}v \} $$ These nonlinear systems involving p-Laplacian appear in many problems in pure and applied mathematics e.g. in quasiconformal mappings, non-Newtonian fluids, and nonlinear elasticity \cite{a3,a4,d1}. The existence of solutions for such systems was proved, using the method of sub and super solutions in \cite{b3,b4,s1}. Here, we use another technique for proving the existence of weak solutions. We use the theory of monotone operators. First, we consider the following system defined on a bounded domain $\Omega$ of $\mathbb{R}^N$ with boundary $\p\Omega$: \begin{gather*} -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f(x),\quad\text{in } \Omega\\ -\Delta_qv=d(x)|v|^{q-2}v-c(x)|v|^\beta|u|^\alpha u + g(x), \quad \text{in }\Omega\\ u=v=0,\quad\text{on }\p \Omega\,. \end{gather*} Then, we generalize the discussion to system defined on the whole space $\mathbb{R}^N$. This article is organized as follows: Some technical results and definitions are introduced in section two concerning the theory of nonlinear monotone operators, also, the scalar case is discussed. Section three, is devoted to study the existence of solutions for nonlinear systems defined on a bounded domain. In section four, the existence of solutions for nonlinear systems defined on unbounded domain is proved. \section{Scalar case} First, we introduce some technical results \cite{b2,b4,z1}. \subsection*{Definitions} Let $A: V \to V'$ be an operator on a Banach space $V$. We say that the operator $A$ is: \\ \emph{Coercive} if $\lim_{\|u\|\to\infty}\frac{\langle A(u),u\rangle}{\|u\|}=\infty$;\\ \emph{Monotone} if $\langle A(u_1)-A(u_2),u_1-u_2\rangle \ge 0$ for all $u_1, u_2$; \\ \emph{Strongly continuous} if $u_n \overset{w}{\to} u$ implies $A(u_n) \to A(u)$; \\ \emph{Weakly continuous} if $u_n \overset{w}{\to} u$ implies $A(u_n) \overset{w}{\to} A(u)$;\\ \emph{Demicontinuous} if $u_n \to u$ implies $A(u_n) \overset{w}{\to} A(u)$.\\ The operator $A$ is said to satisfy the $M_o$-condition if $u_n\overset{w}\to u$, $A(u_n) \overset{w}{\to} f$, and $[\langle A(u_n),u_n\rangle \to \langle f,u \rangle]$ imply $A(u)=f$. \begin{theorem} \label{thm1} Let $V$ be a separable reflexive Banach space and $A: V\to V'$ an operator which is: coercive, bounded, demicontinuous, and satisfying $M_o$ condition. Then the equation $A(u)=f$ admits a solution for each $f\in V'$. \end{theorem} Now, we prove the existence of a weak solution $u\in W^{1,p}_{0}(\Omega)$ for the scalar case \begin{equation} \label{T} \begin{gathered} -\Delta_p u= m(x)|u|^{p-2}u+f(x), \quad x \in \Omega,\\ u=0\qquad\text{on } \p\Omega \end{gathered} \end{equation} where $01$ . First, we prove that $A$ is a bounded operator: $$ (Au, v)=\int_\Omega|\nabla u|^{p-2}\nabla u \nabla v - \int_\Omega m(x) |u|^{p-2} uv $$ Using H\"older's inequality, we obtain \begin{align*} |(Au, v)| & \le \Big(\int_\Omega|\nabla u|^p\Big)^{\frac{p-1}{p}}\Big(\int_\Omega|\nabla v|^p\Big)^{\frac{1}{p}}+ \Big(\int_\Omega m(x)| u|^p\Big)^{\frac{p-1}{p}}\Big(\int_\Omega m(x)| v|^p\Big)^{\frac{1}{p}}\\ &\le \|u\|^{p-1}_{1,p}\|v\|_{1,p} \end{align*} To prove that $A$ is continuous, let us assume that $u_n\to u$ in $W^{1,p}_{0}(\Omega)$. Then $\|u_n-u\|_{1, p}\to 0$ So that $$ \|\nabla u_n-\nabla u\|_p\to 0 $$ Applying Dominated convergence theorem, we obtain $$ \| |\nabla u_n|^{p-2} \nabla u_n -|\nabla u|^{p-2} \nabla u\|_{p}\to 0 $$ hence $$ \|Au_n - Au\|_{p}\le \||\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u \|_{p}+ \||u_n|^{p-2}u_n-|u|^{p-2}u \|_{p}\to 0 $$ Operator $A$ is strictly monotone: \begin{align*} ( Au_1-Au_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\\ &\ge \int_\Omega|\nabla u_1|^{p}+\int_\Omega\!|\nabla u_2|^{p} -\Big(\int_\Omega|\nabla u_1|^{p}\Big)^{p-1/p}\Big(\int_\Omega|\nabla u_2|^{p}\Big)^{1/p}\\ &\quad -\Big(\int_\Omega|\nabla u_2|^{p}\Big)^{p-1/p}\Big(\int_\Omega|\nabla u_1|^{p}\Big)^{1/p}\\ &=\|u_1\|^{p}_{p}+\|u_2\|^{p}_{p}-\|u_1\|^{p-1}_{p}\|u_2\|_{p} -\|u_2\|^{p-1}_{p}\|u_1\|_{p}\\ &=\big(\|u_1\|^{p-1}_{1,p}-\|u_2\|^{p-1}_{1,p}\big) \big(\|u_1\|_{1,p}-\|u_2\|_{1,p}\big)>0\,. \end{align*} Also, $A$ is a coercive operator, since from \eqref{e1}, we have \begin{align*} ( Au , u)&=\int_\Omega|\nabla u|^p- \int_\Omega m| u|^p\\ &\ge\int_\Omega|\nabla u|^p-\frac{1}{\lambda_p(m)}\int_\Omega|\nabla u|^p\\ &=\Big(1-\frac{1}{\lambda_p(m)}\Big)\int_\Omega|\nabla u|^p\,. \end{align*} Then $$ \frac{( Au, u)}{\|u\|_{p}}=\|u\|^{p-1}_{1,p}\to \infty\quad\text{as}\quad \|u\|_{1,p}\to \infty $$ which proves the existence of a weak solution for \eqref{T}. %\end{proof} \section{Nonlinear systems on bounded domains} In this section, we consider the system \begin{equation} \label{P} \begin{gathered} -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f(x),\quad \text{in }\Omega\\ -\Delta_qv=d(x)|v|^{q-2}v-c(x)|v|^\beta|u|^\alpha u + g(x), \quad \text{in }\Omega\\ u=v=0,\quad\text{on }\p \Omega \end{gathered} \end{equation} where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $\frac{1}{p}+\frac{1}{p'}=1,\q \frac{1}{q}+\frac{1}{q'}=1$, $\alpha+\beta+21,\q\text {and}\q \lambda_q(d)>1\,. \end{equation} \end{theorem} \begin{proof} We transform the weak formulation of the system \eqref{P} to the operator form $$ A(u,v)-B(u,v)=F $$ where, $A, B$ and $F$ are operators defined on $W^{1,p}_{0}(\Omega)\times W^{1,q}_{0}(\Omega)$ by $$ ( A(u,v), (\Phi_1,\Phi_2))=\int_\Omega|\nabla u|^{p-2}\nabla u \nabla \Phi_1+ \int_\Omega|\nabla v|^{q-2}\nabla v \nabla \Phi_2, $$ \begin{align*} (B(u,v), (\Phi_1,\Phi_2))=&\int_\Omega a(x) |u|^{p-2}u\Phi_1+\int_\Omega d(x) |v|^{q-2}v \Phi_2\\ &- \int_\Omega b(x) |u|^\alpha |v|^\beta v \Phi_1- \int_\Omega c (x) |v|^\beta |u|^\alpha u \Phi_2 \end{align*} and $$ (F,\Phi)=((f_1, f_2),(\Phi_1,\Phi_2))=\int_\Omega f_1 \Phi_1 +\int_\Omega f_2 \Phi_2 $$ We can write the operator $A(u,v)$ as the sum of the two operators $J_2(v), J_1(u)$, where $$ ( J_2 (v),(\Phi_2))=\int_\Omega|\nabla v|^{q-2}\nabla v \nabla \Phi_2\quad\text{and}\quad ( J_1(u),(\Phi_1))=\int_\Omega|\nabla u|^{p-2}\nabla u \nabla \Phi_1\,. $$ Operators $J_1$ and $J_2$ are bounded, continuous, and strictly monotone; so their sum, the operator $A$, will be the same. For the operator $B(u,v)$, $$ B(u,v):W^{1,p}_{0}(\Omega)\times W^{1,q}_{0}(\Omega)\to L^{p}(\Omega)\times L^{q}(\Omega)\subset W^{-1,p'}_{0}(\Omega)\times W^{-1,q'}_{0}(\Omega), $$ using Dominated convergence theorem and compact imbedding property \cite{a1} for the space $W^{1,p}_{0}(\Omega)$ inside the space $L^p(\Omega)$ and the space $W^{1,q}_{0}(\Omega)$ inside $L^q(\Omega)$, when $\Omega$ is a bounded domain of $\mathbb{R}^N$, we can prove that it is a strongly continuous operator. To prove that let us assume that $v_n\to^{\hskip-0.3cm w} v $ in $W^{1,q}_{0}(\Omega)$ and $u_n\overset{w}{\to} u$ in $W^{1,p}_{0}(\Omega)$. Then $(u_n, v_n)\to (u,v)$ in $L^p(\Omega)\times L^q(\Omega)$. Also, $(\nabla u_n, \nabla v_n)\to(\nabla u, \nabla v)$ in $L^p(\Omega)\times L^q(\Omega)$. By the Dominated Convergence Theorem, we have: \begin{gather*} a(x)|u_n|^{p-2}u_n\to a(x)|u|^{p-2}u\quad\quad\quad\text{in } L^{p}(\Omega)\\ d(x)|v_n|^{q-2}v_n\to d(x)|v|^{q-2}v\quad\quad\quad\text{in } L^{q}(\Omega)\\ -b(x)|u_n|^\alpha|v_n|^\beta v_n\to -b(x)|u|^\alpha|v|^\beta v \q\quad\text{in } L^{p}(\Omega)\\ -c(x)|v_n|^\beta|u_n|^\alpha u_n\to -c(x)|v|^\beta|u|^\alpha u \q\quad\text{in } L^{q}(\Omega)\,. \end{gather*} Since \begin{align*} &( B(u_n,v_n)-B(u,v), (w_1, w_2))\\ &=\int_{\Omega}a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)w_1 +\int_{\Omega}d(x)(|v_n|^{q-2}v_n-|v|^{q-2}v)w_2\\ &- \int b(x)(|u_n|^\alpha|v_n|^\beta v_n-|u|^\alpha|v|^\beta v)w_1 -\int_{\Omega} c(x)(|v_n|^\beta|u_n|^\alpha u_n-|v|^\beta|u|^\alpha u)w_2, \end{align*} it follows that \begin{align*} &\|B(u_n,v_n)-B(u,v)\| \\ &\le\|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{p} +\|d(x)(|v_n|^{q-2}v_n-|v|^{q-2}v)\|_{q}\\ &+\|b(x)(|u_n|^\alpha|v_n|^{\beta+1} -|u|^\alpha|v|^{\beta+1} )\|_{p} +\| c(x)(|u_n|^{\alpha+1}|v_n|^\beta -|u|^{\alpha+1}|v|^\beta)\|_{q}\to 0\,. \end{align*} This proves that $-B(u,v)$ is a strongly continuous operator. So $A(u,v)-B(u,v)$ will be an operator satisfying the $M_o$-condition. Now, it remains to prove that $A(u,v)-B(u,v)$ is a coercive operator: \begin{align*} &|(A(u,v)-B(u,v),(u,v))| \\ &=\int_\Omega|\nabla u|^p+ \int|\nabla v|^q-\int_\Omega a(x)|u|^p-\int d(x)|v|^q\\ &\quad +\int_\Omega b(x)|u|^{\alpha+1}|v|^{\beta+1}+\int_\Omega c(x)|u|^{\alpha+1}|v|^{\beta+1}\\ &\ge\int_\Omega|\nabla u|^p+\int_\Omega|\nabla v|^q-\frac{1}{\lambda_p(a)}\int_\Omega|\nabla u|^p-\frac{1}{\lambda_q(d)} \int_\Omega|\nabla v|^q\\ &=\Big(1-\frac{1}{\lambda_p(a)}\Big)\int_\Omega|\nabla u|^p+ \Big(1-\frac{1}{\lambda_q(d)}\Big)\int_\Omega|\nabla v|^q \end{align*} From \eqref{e2}, we deduce $$ (A(u,v)-B(u,v), (u,v))\ge c (\|u\|^{p}_{1,p}+\|v\|^{q}_{1,q})=c|(u,v)\|_{W^{1,p}_{0}\times W^{1,q}_{0}} $$ So that $$ \langle A(u,v)-B(u,v),(u,v)\rangle\to\infty\quad\text{as}\quad \|(u,v)\|_{W^{1,p}_{0}\times W^{1,q}_{0}}\to\infty\,. $$ This proves the coercive condition and so, the existence of a weak solution for system \eqref{P}. \end{proof} \section{Nonlinear systems defined on $\mathbb{R}^n$} We consider the nonlinear system \begin{equation} \label{S} \begin{gathered} -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f,\\ -\Delta_qv=-c(x)|u|^\alpha |v|^\beta u + d(x) |v|^{q-2}v +g ,\\ \lim_{|x|\to\infty}u=\lim_{|x|\to\infty}v=0\quad u,v>0 \end{gathered} \end{equation} which is defined on $\mathbb{R}^N$. We assume that $1\le \frac{2N}{N+1}0$ such that for all $u\in D^{1,p}(\mathbb{R}^N)$, \begin{equation} \label{e5} \|u\|_{L^{Np/(N-p)}}\le K \|u\|_{D^{1,p}(\mathbb{R}^N)}\,. \end{equation} Clearly, the space $D^{1,p}(\mathbb{R}^N)$ is a reflexive Banach space embedded continuously in the space $L^{Np/(N-p)}(\mathbb{R}^N)$. \begin{lemma} \label{lem1} The eigenvalue problem \begin{equation} \label{sigma} \begin{gathered} -\Delta_p u=\lambda a(x)|u|^{p-2}u\quad \text{in } \mathbb{R}^N\\ u(x)\to 0\quad\text{as } |x|\to\infty \end{gathered} \end{equation} admits a positive principal eigenvalue $\Lambda_a(p)$ which is associated with a positive eigenfunction $\phi\in D^{1,p}(\mathbb{R}^N)$; moreover $\Lambda_a(p)$ is characterized by \begin{equation} \label{e6} \Lambda_a(p)\int_{R^N} a(x)|u|^p\le \int_{R^N}|\nabla u|^p, \quad\forall u\in D^{1,p}(\mathbb{R}^N) \end{equation} \end{lemma} \begin{theorem} \label{thm3} For $(f,g)\in L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N)\times L^{\frac{Nq}{N(q-1)+q}}(\mathbb{R}^N)$, there exists a weak solution $(u,\,v)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ for system \eqref{S} if the following conditions are satisfied: \begin{equation} \label{e7} \Lambda_p(a)>1,\quad\text{and}\quad \Lambda_q(d)>1\,. \end{equation} \end{theorem} \begin{proof} By transforming the weak formulation for the system to the operator formulation, we will get the bounded operators $A, B, F$ on the space $D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ which take the same previous definitions in Theorem \ref{thm2}. To distinguish that: let us assume that $(\Phi_1, \Phi_2)$ in $D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$, then applying H\"older inequality, we get \begin{align*} &|( A(u,v), (\Phi_1, \Phi_2))| \\ & \le \int_{R^N} |\nabla u|^{p-1}|\nabla \Phi_1|+\int_{R^N}|\nabla v|^{q-1}|\nabla \Phi_2|\\ &\le \Big(\int_{R^N}|\nabla u|^{p}\Big)^{\frac{p-1}{p}} \Big(\int_{R^N}|\nabla \Phi_1|^p\Big)^{\frac{1}{p}}+ \Big(\int_{R^N}|\nabla v|^{q}\Big)^{\frac{q-1}{q}} \Big(\int_{R^N}| \nabla \Phi_2|^q\Big)^{\frac{1}{q}}\\ &=\|u\|^{p-1}_{D^{1,p}}\|\Phi_1\|_{D^{1,p}}+\|v\|^{q-1}_{D^{1,q}} \|\Phi_2\|_{D^{1,q}}\\ &\le(\|u\|^{p-1}_{D^{1,p}}+\|v\|^{q-1}_{D^{1,q}}) (\|\Phi_1\|_{D^{1,p}}+\|\Phi_2\|_{D^{1,q}})\\ &=\big(\|u\|^{p-1}_{D^{1,p}}+\|v\|^{q-1}_{D^{1,q}}\big) \|(\Phi_1,\Phi_2)\|_{D^{1,p}\times D^{1,q}} \end{align*} For the operator $B(u,v)$, we have \begin{align*} & |( B(u,v),(\Phi_1,\Phi_2))|\\ &\le \Big(\int (a(x))^{\frac{N}{p}}\Big)^{\frac{p}{N}} \Big(\int_{R^N}|u(x)|^{\frac{Np}{N-p}}\Big)^{\frac{(p-1)(N-p)}{Np}} \Big(\int_{R^N}|\Phi_1|^{\frac{Np}{N-p}}\Big)^{\frac{N-p}{Np}} \\ &\quad +\Big(\int_{R^N}(d(x))^{\frac{N}{q}}\Big)^{\frac{q}{N}} \Big(\int_{R^N}|v|^{\frac{Nq}{N-q}}\Big)^{\frac{(q-1)(N-q)}{Nq}} \Big(\int_{R^N}|\Phi_2|^{\frac{Nq}{N-q}}\Big)^{\frac{N-q}{Nq}}\\ &\quad +\Big(\int_{R^N}(b(x))^{\frac{N}{\alpha+\beta+2}}\Big) ^{\frac{\alpha+\beta+2}{N}} \Big(\int_{R^N}|u|^{\frac{Np}{N-p}}\Big)^{\frac{\alpha(N-p)}{Np}} \Big(\int_{R^N}|v|^{\frac{Nq}{N-q}}\Big)^{\frac{(\beta+1)(N-q)}{Nq}}\\ &\quad \times \Big(\int_{R^N}|\Phi_1|^{\frac{Np}{N-p}}\Big)^{\frac{N-p}{Np}} +\Big(\int_{R^N}(c(x))^{\frac{N}{\alpha+\beta+2}}\Big)^{\frac{\alpha+\beta+2}{N}} \Big(\int_{R^N}|u|^{\frac{Np}{N-p}}\Big)^{\frac{(\alpha+1)(N-p)}{Np}}\\ &\quad\times \Big(\int_{R^N}|v|^{\frac{Nq}{N-q}}\Big)^{\frac{\beta(N-q)}{Nq}} \Big(\int_{R^N}|\Phi_2|^{\frac{Np}{N-p}}\Big)^{\frac{N-q}{Nq}}\\ &\le k_1\|u\|^{p-1}_{D^{1,p}}\|\Phi_1\|_{D^{1,p}}+k_2\|v\|^{q-1}_{D^{1,q}} \|\Phi_2\|_{D^{1,q}}\\ &\quad +k_3\|u\|^{\alpha}_{D^{1,p}}\|v\|^{\beta+1}_{D^{1,p}} \|\Phi_1\|_{D^{1,p}}+ k_4\|u\|^{\alpha+1}_{D^{1,p}}\|v\|^{\beta}_{D^{1,q}}\|\Phi_2\|_{D^{1,q}} \\ &\le \Big(k_1\|u\|^{p-1}_{D^{1,p}}+k_2\|v\|^{q-1}_{D^{1,q}}+k_3\|u\| ^{\alpha}_{D^{1,p}}\|v\|^{\beta+1}_{D^{1,p}} +k_4\|u\|^{\alpha+1}_{D^{1,p}}\|v\|^{\beta}_{D^{1,q}}\Big)\\ &\quad\times \|(\Phi_1,\Phi_2)\|_{D^{1,p}\times D^{1,q}}, \end{align*} this proves the boundedness of the operator $B(u,v)$. For $F$, we have \begin{align*} |(F,\Phi)| &=|((f_1,f_2), (\Phi_1, \Phi_2))|\\ &\le \Big(\int_{R^N}(|f_1|)^{\frac{Np}{N(p-1)+p}}\Big)^{\frac{N(p-1)+p}{Np}} \Big(\int_{R^N}|\Phi_1|^{\frac{Np}{N-p}}\big)^{\frac{N-p}{Np}}\\ &\quad +\Big(\int_{R^N}(|f_2|)^{\frac{Nq}{N(q-1)+q}}\Big)^{\frac{N(q-1)+q}{Nq}} \Big(\int_{R^N}|\Phi_2|^{\frac{Nq}{N-q}}\Big)^{\frac{N-q}{Nq}} \\ &\le\Big( \|f_1\|_{\frac{Np}{N(p-1)+p}}+ \|f_2\|_{\frac{Nq}{N(q-1)+q}}\Big) \|(\Phi_1,\Phi_2)\|_{D^{1,p}\times D^{1,q}}. \end{align*} Now, the operator $A(u,v)=J_1(u)+J_2(v)$ is continuous and strictly monotone on $D^{1,p}\times D^{1,q}$, since \begin{align*} ( J_1(u_1)-J_1(u_2),u_1-u_2)\ge (\|u_1\|^{p-1}_{D^{1,p}}-\|u_2\|^{p-1}_{D^{1,p}}) (\|u_1\|_{D^{1,p}}-\|u_2\|_{D^{1,q}})>0, \\ ( J_2(u_1)-J_2(u_2),u_1-u_2)\ge (\|u_1\|^{q-1}_{D^{1,q}}-\|u_2\|^{q-1}_{D^{1,q}}) (\|u_1\|_{D^{1,q}}-\|u_2\|_{D^{1,q}})>0 \end{align*} For the operator $B(u,v)$, we can prove that it is a strongly continuous operator by using Dominated convergence theorem and continuous imbedding property for the space $D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ into $L^{\frac{Np}{N-p}}(\mathbb{R}^N)\times L^{\frac{Nq}{N-q}}(\mathbb{R}^N)$: let us assume that $v_n\to^{\hskip-0.3cm w} v$ in $D^{1,q}(\mathbb{R}^N)$ and $u_n\to^{\hskip-0.3cm w} u$ in $D^{1,p}(\mathbb{R}^N)$. Then $(u_n, v_n)\to (u,v)$ in $L^p(\mathbb{R}^N)\times L^q(\mathbb{R}^N)$ and $(\nabla u_n, \nabla v_n)\to (\nabla u,\nabla v)$ in $L^p(\mathbb{R}^N)\times L^q(\mathbb{R}^N)$. Now, the sequence $(u_n)$ is bounded in $D^{1,p}(\mathbb{R}^N)$, then it is containing a subsequence again denoted by $(u_n)$ converges strongly to $u$ in $L^{\frac{Np}{N-p}}(B_{r_0})$ for any bounded ball $B_{r_0}=\{x\in \mathbb{R}^N:\|x\|\le r_0\}$ . Similarly $(v_n)$ converges strongly to $v$ in $L^{\frac{Nq}{N-q}}(B_{r_0})$. Since $u_n, u \in L^{\frac{Np}{N-p}}(B_{r_0})$ and $v_n, v \in L^{\frac{Nq}{N-q}}(B_{r_0})$. Then using the dominated convergence theorem, we have \begin{gather} \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{\frac{Np}{N(p-1)+p}}\to 0,\label{e8}\\ \|d(x)(|v_n|^{q-2}v_n-|v|^{q-2}v)\|_{\frac{Nq}{N(q-1)+q}}\to 0,\label{e9}\\ \|b(x) (|u_n|^{\alpha-1}|v_n|^{\beta+1}u_n- |u|^{\alpha-1}|v|^{\beta+1}u)\|_{\frac{Np}{N(p-1)+p}}\to 0,\label{e10}\\ \|c(x) (|u_n|^{\alpha+1}|v_n|^{\beta-1}u_n- |u|^{\alpha+1}|v|^{\beta-1}u)\|_{\frac{Nq}{N(q-1)+q}}\to 0\,.\label{e11} \end{gather} Then \begin{align*} & \|B(u_n,v_n)-B(u,v)\|_{D^{1,p}(B_{r_0})\times D^{1,q}(B_{r_0})} \\ &\le \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{\frac{Np}{N(p-1+p)}} +\|d(x)(|v_n|^{q-2}v_n-|v|^{q-2}v)\|_{\frac{Nq}{N(q-1+q)}}\\ &\quad +\|b(x)(|u_n|^\alpha |v_n|^{\beta+1}u_n-|u|^\alpha|v|^{\beta+1}v)\|_{\frac{Np}{N(p-1)+p}}\\ &\quad +\|c(x)(|u_n|^{\alpha+1} |v_n|^{\beta-1}u_n-|u|^{\alpha+1}|v|^{\beta-1}v)\|_{\frac{Nq}{N(q-1)+q}}\to 0\,. \end{align*} It remains to study the norm $$ \|B(u_n,v_n)-B(u,v)\|_{D^{1,p}(\mathbb{R}^N-B_{r_0})\times D^{1,q}(\mathbb{R}^N-B_{r_0})} $$ It is sufficient to study the norms in the inequalities \eqref{e8}--\eqref{e11} and try to make it as small as possible. We will study the norm in \eqref{e8} only because the others will be the same. Since, $(u_n)$ converges weakly in the space $D^{1,p}(\mathbb{R}^N)$, using Sobelev inequality, $(u_n)$ will be bounded in the space $L^{\frac{Np}{N-p}}(\mathbb{R}^N)$, so $|u_n|^{p-1}$ will be bounded in $L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N-B_{r_0})$ and $(|u_n|^{p-2}u_n-|u|^{p-2}u)$ is bounded in $L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N-B_{r_0})$. Since, $a(x) \in L^{\frac{N}{p}}(\mathbb{R}^N)$, we can make the integral $\int_{(\mathbb{R}^N-B_{r_0})}|a(x)|^{\frac{N}{p}}$ as small as possible by choosing $r_0$ big as possible, this means that there exists $r_0>0$ such that $$ \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{L^{\frac{Np}{N(p-1)+p}} (\mathbb{R}^N-B_{r_0})} < \frac{\epsilon }{4}. M = \frac{\epsilon }{4} $$ for all $n \ge N_0$, $r\ge r_0$. Since \begin{align*} &\|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{L^{\frac{Np}{N(p-1)+p}}(\mathbb{R}^N)}\\ &= \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{L^{\frac{Np}{N(p-1)+p}}(B_{r_0})}\\ &\quad + \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{L^{\frac{Np}{N(p-1)+p}} (\mathbb{R}^N-B_{r_0})}, \end{align*} it follows that $$ \|a(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)\|_{L^{\frac{Np}{N(p-1)+p}} (\mathbb{R}^N)}\to 0\,. $$ By repeating the previous steps on the remaining terms in $$ \|B(u_n,v_n)-B(u,v)\|_{D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N) }, $$ we can prove that this norm tending strongly to zero and then the operator $B(u,v)$ is strongly continuous. It remains to justify that the operator $A(u,v)-B(u,v)$ is a coercive operator. From \eqref{e4}, \eqref{e6} and \eqref{e7}, we obtain \begin{align*} &( A(u,v)-B(u,v),(u,v)) \\ &=\int_{\mathbb{R}^N}|\nabla u|^p+\int_{\mathbb{R}^N}|\nabla v|^q -\int_{\mathbb{R}^N} a(x) |u|^p- \int_{\mathbb{R}^N} d(x) |v|^q\\ &\quad +\int b(x)|u|^{\alpha+1}|v|^{\beta+1}+\int b(x)|u|^{\alpha+1}|v|^{\beta+1}\\ &\ge \int_{\mathbb{R}^N}|\nabla u|^p+\int_{\mathbb{R}^N}|\nabla v|^q -\frac{1}{\Lambda_a(p)}\int_{\mathbb{R}^N} |\nabla u|^p- \frac{1}{\Lambda_d(q)}\int_{\mathbb{R}^N} |\nabla v|^q\\ &=\Big(1-\frac{1}{\Lambda_a(p)}\Big)\int_{\mathbb{R}^N} |\nabla u|^p+ \Big(1-\frac{1}{\Lambda_d(q)}\Big)\int_{\mathbb{R}^N} |\nabla v|^q\\ &> c\Big(\|u\|^{p}_{D^{1,p}}+\|v\|^{q}_{D^{1,q}}\Big)\,. \end{align*} So that $$ ( A(u,v)-B(u,v), (u,v)) \to \infty \quad \text{as}\quad \|(u,v)\|_{D^{1.p}\times D^{1,q}}\to \infty $$ The coercive condition for the operator completes the proof of the existence of a weak solution for system \eqref{S}. \end{proof} \begin{thebibliography}{00} \bibitem{a1} R. A. 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