\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 82, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/82\hfil Maximum principle and existence results] {Maximum principle and existence results for nonlinear cooperative systems on a bounded domain} \author[L. Leadi, A. Marcos\hfil EJDE-2011/82\hfilneg] {Liamidi Leadi, Aboubacar Marcos} % in alphabetical order \address{Liamidi Leadi \newline Institut de Math\'ematiques et de Sciences Physiques, Universit\'e d'Abomey Calavi, 01 BP: 613 Porto-Novo, B\'enin (West Africa)} \email{leadiare@imsp-uac.org, leadiare@yahoo.com} \address{Aboubacar Marcos \newline Institut de Math\'ematiques et de Sciences Physiques, Universit\'e d'Abomey Calavi, 01 BP: 613 Porto-Novo, B\'enin (West Africa)} \email{abmarcos@imsp-uac.org} \thanks{Submitted November 11, 2010. Published June 24, 2011.} \subjclass[2000]{35B50, 35J57, 35J60} \keywords{Maximum Principle; elliptic systems; p-Laplacian operator; \hfill\break\indent sub-super solutions; approximation method} \begin{abstract} In this work we give necessary and sufficient conditions for having a maximum principle for cooperative elliptic systems involving $p$-Laplacian operator on a bounded domain. This principle is then used to yield solvability for the considered cooperative elliptic systems by an approximation method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} This article studies the general nonlinear cooperative elliptic system \begin{equation} \label{eS} \begin{gathered} -\Delta_pu = am(x)|u|^{p-2}u + bm_1(x)h(u,v) + f \quad\text{in }\Omega\\ -\Delta_qv = dn(x)|v|^{q-2}v + cn_1(x)k(u,v) + g \quad\text{in }\Omega\\ u = v = 0 \quad\text{on }\partial{\Omega} \end{gathered} \end{equation} where $\Omega$ is an bounded domain of class $C^{2, \nu}$ of $\mathbb{R}^N$ $(N\geq 1)$. Here $\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$, $1 1$ and $\frac{\alpha + 1}{p} + \frac{\beta + 1}{q} = 1$; \item[(B2)] $b,c\geq 0$, $f\in L^{p'}(\Omega)$, $g\in L^{q'}(\Omega)$ with $\frac{1}{p} + \frac{1}{p'} = \frac{1}{q} + \frac{1}{q'} = 1$; \item[(B3)] $ m, m_1,n, n_1$ are smooth weights such that $m,n\in L^{\infty}(\Omega)$ and $0 < m_1$, $n_1 \leq m^{(\alpha + 1)/p} n^{(\beta +1)/q}$. \item[(B4)] The functions $h$ and $k$ satisfy the sign conditions: $th(s,t)\geq 0$, $s k(s,t)\geq 0$ for $(s,t)\in \mathbb{R}^2$ and there exits $\Gamma > 0$ such that \begin{gather*} h(s,-t) \leq - h(s,t) \quad \text{for } t \geq 0,\; s \in \mathbb{R}\\ h(s,t) = \Gamma^{\alpha +\beta + 2 - p }|s|^{\alpha}|t|^{\beta}t \quad \text{for } t \leq 0,\; s\in \mathbb{R} \end{gather*} and \begin{gather*} k(-s,t) \leq - k(s,t) \quad \text{for } s \geq 0 , t\in \mathbb{R}\cr k(s,t) = \Gamma^{\alpha +\beta + 2 - q }|s|^{\alpha}s|t|^{\beta} \quad \text{for } s \leq 0, t \in \mathbb{R} \end{gather*} \end{itemize} Here and henceforth the Lebesgue norm in $L^p(\Omega)$ will be denoted by $\|\cdot\|_p$ and the usual norm of $W_0^{1,p}(\Omega)$ by $\|\cdot\|$. The positive and negative part of a function $u$ are defined respectively as $u^+:=\max\{u,0\}$ and $u^-:=\max\{-u,0\}$. Equalities (and inequalities) between two functions must be understood a.e. in $\Omega$. Let us recall some results on eigenvalue problems with weight (cf \cite{ana,all}) useful in the sequel for this work. Given $ g \in L^{\infty}(\Omega)$, it was known that the eigenvalue problem \begin{equation} \label{e2.1} \begin{gathered} - \Delta_p u = \lambda g(x)|u|^{p-2}u \quad\text{in }\Omega\\ u = 0 \quad \text{on }\partial{\Omega} \end{gathered} \end{equation} admits, an unique positive first eigenvalue $\lambda_1(g,p)$ with a nonnegative eigenfunction. Moreover, this eigenvalue is isolated, simple and as a consequence of its variational characterization one has $$ {\lambda_1(g,p)\int_\Omega g(x)|u|^p \leq \int_\Omega |\nabla u|^p\quad \forall u\in W_0^{1,p}(\Omega)}. $$ Now we denote by $\Phi$ (respectively $\Psi$) the positive eigenfunction associated with $\lambda_1(m,p)$ (respectively $\lambda_1(n,q)$) normalized by $\int_{\Omega}m(x)|\Phi|^{p} = 1$ (resp $\int_{\Omega}n(x)|\Psi|^{q} = 1 )$. The functions $\phi$ and $\psi$ belong to $C^{1, \alpha}(\bar{\Omega})$ (see \cite{ser,tolk}) and by the weak maximum principle (see \cite{vaz}) $$ \frac{\partial{\Phi}}{\partial{\nu}} < 0 \quad\text{and}\quad \frac{\partial{\Psi}}{\partial{\nu}} < 0 \quad\text{on } \partial{\Omega}, $$ where $\nu$ is the unit exterior normal. Finally, let us define $$ \Theta:={\frac{{\inf_{\Omega}}k_1(x)} {{\sup_{\Omega}}k_2(x)}}, $$ where \[ k_1(x):=[\frac{n_1(x)}{n(x)}]^{(\beta +1)/q} [\frac{\Phi(x)^p}{\Psi(x)^q}]^{\frac{\alpha +1}{p} \frac{\beta +1}{q}}, \quad k_2(x):=[\frac{m(x)}{m_1(x)}]^{(\alpha +1)/p} [\frac{\Phi(x)^p}{\Psi(x)^q}]^{\frac{\alpha +1}{p}\frac{\beta +1}{q}}. \] \section{A Maximum Principle for system \eqref{eS}} We say that a Maximum Principle holds for system \eqref{eS} if $f\geq 0$ and $g\geq 0$ implies $u\geq 0$ and $v\geq 0$. By a solution $(u,v)$ of \eqref{eS}, we mean a weak solution; i.e., $(u,v)\in W_{0} ^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$ such that \begin{equation}\label{a2`} \begin{gathered} {\int_{\Omega}|\nabla u|^{p-2}\nabla u.\nabla w= \int_{\Omega} [am(x)|u|^{p-2}uw+bm_1(x)h(u,v)w+fw]} \\ {\int_{\Omega}|\nabla v|^{q-2}\nabla v.\nabla z= \int_{\Omega} [dn(x)|v|^{q-2} vz + cn_1(x)k(u,v)z+ gz]} \end{gathered} \end{equation} for all $(w,z)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$. Note that by assumptions (B1)--(B4), the integrals in \eqref{a2`} are well-defined. We are now ready to state the validity of the Maximum Principle for \eqref{eS}. \begin{theorem} \label{thm1} Assume (B1)--(B4). Then the Maximum Principle holds for \eqref{eS} if %\label{a2} \begin{itemize} \item[(C1)] $\lambda_1(m,p) > a$, \item[(C2)] $\lambda_1(n,q) > d$, \item[(C3)] $(\lambda_1(m,p) - a )^{(\alpha + 1)/p} (\lambda_1(n,q)- d )^{(\beta + 1)/q} > b^{(\alpha + 1)/p}c{^{(\beta + 1)/q}}$. \end{itemize} Conversely if the Maximum Principle holds, then conditions {\rm (C1)--(C4)} are satisfied, where \begin{itemize} \item[(C4)] $(\lambda_1(m,p) - a )^{(\alpha + 1)/p} (\lambda_1(n,q)- d )^{(\beta + 1)/q} > \Theta b^{(\alpha + 1)/p}c{^{(\beta + 1)/q}}$ \end{itemize} \end{theorem} \begin{proof} The proof is partly adapted from \cite{bouch,lea} \textbf{The condition is necessary.} Assume that the Maximum Principle holds for system \eqref{eS}. If $\lambda_1(m,p)\leq a$ then the functions $f:=(a - \lambda_1(m,p))m(x)\Phi^{p-1}$ and $ g:= 0$ are nonnegative, however $(-\Phi , 0)$ satisfies \eqref{eS}, which contradicts the Maximum Principle. Similarly, if $\lambda_1(n,q)\leq d$ then $f:= 0 $ and $g:=(d - \lambda_1(n,q) )n(x)\Psi^{q-1}$ are nonnegative functions and $(0 , -\Psi)$ satisfies \eqref{eS}, which is a contradiction with the Maximum Principle. Now, assume that $\lambda_1(m,p) > a$, $\lambda_1(n,q) > d$ and that (C4) does not hold; that is, \begin{itemize} \item[(C4')] $(\lambda_1(m,p) - a )^{(\alpha + 1)/p} (\lambda_1(n,q)- d )^{(\beta + 1)/q} < \Theta b^{(\alpha + 1)/p}c{^{(\beta + 1)/q}}$ \end{itemize} Set $$ A=\Big(\frac{\lambda_1(m,p)-a}{b}\Big)^{(\alpha + 1)/p}, \quad B =\Big(\frac{\lambda_1(n,q)-d}{c}\Big)^{(\beta + 1)/q}, $$ then (C4') becomes $AB \leq \Theta $ which implies \begin{equation} \label{a'2} A\Theta_2\leq \frac{\Theta_1}{B}, \quad \text{where } \Theta_1= {\inf_\Omega}k_1(x), \quad \Theta_2={\sup_\Omega}k_2(x). \end{equation} Hence there exists $\xi>0$ such that $$ A\Theta_2\leq \xi\leq \frac{\Theta_1}{B}. $$ Let $c_1,c_2$ be two positive real numbers such that $$ \xi = \Big(\frac{c_2^q \Gamma^q}{c_1^p \Gamma^p}\Big) ^{\frac{\alpha+1}{p} \frac{\beta +1}{q}}. $$ Using \eqref{a'2}, (B1) and the above expression of $\xi$, we have $$ [\lambda_1(m,p)-a]m(x)[c_1\Phi(x)]^{p-1}\leq \Gamma^{\alpha +\beta +2 -p}bm_1(x)[c_1\Phi(x)] ^{\alpha}[c_2\Psi(x)]^{\beta +1} $$ a.e, for $x\in \Omega$ and $$ [\lambda_1(n,q)-d]n(x)[c_2\Psi(x)]^{q-1} \leq \Gamma^{\alpha +\beta +2 -q}cn_1(x)[c_1\Phi(x)]^{\alpha +1} [c_2\Psi(x)]^{\beta} $$ a.e, for $x\in \Omega$. Furthermore, using the inequalities in (B4), we obtain $$ -[\lambda_1(m,p)-a]m(x)[c_1\Phi(x)]^{p-1}- bm_1(x)h(-c_1\Phi,-c_2\Psi) \geq 0 \quad\text{a.e, for}\quad x\in \Omega$$\\ and $$ -[\lambda_1(n,q)-d]n(x)[c_2\Psi(x)]^{q-1} -cn_1(x)k(-c_1\Phi,-c_2\Psi)\geq 0 \quad\text{a.e, for } x\in \Omega . $$ Hence \begin{gather*} 0\leq -[\lambda_1(m,p)-a]m(x)[c_1\Phi(x)]^{p-1} -bm_1(x)h(-c_1\Phi,-c_2\Psi)= f,\\ 0\leq -[\lambda_1(n,q)-d]n(x)[c_2\Psi(x)]^{q-1} -cn_1(x)k(-c_1\Phi,-c_2\Psi)= g \end{gather*} are nonnegative functions and $(-c_1\Phi,-c_2\Psi)$ is a solution of \eqref{eS}. This is a contradiction with the Maximum Principle. \textbf{The condition is sufficient.} Assume that the conditions (C1)--(C3) are satisfied. So for $f\geq 0$, $g\geq 0$, suppose that there exists a solution $(u, v)$ of system \eqref{eS}. Multipling the first equation in \eqref{eS} by $u^-$ and the second one by $v^-$ and integrating over $\Omega$ we have \begin{gather*} { \int_\Omega |\nabla u^-|^{p} = a\int_\Omega m(x)|u^-|^{p} - b\int_\Omega m_1(x)h(u,v)u^{-} - \int_\Omega fu^-}\\ {\int_\Omega |\nabla v^-|^{q} = d\int_\Omega n(x)|v^-|^{q} - c\int_\Omega n_1(x)k(u,v)v^- - \int_\Omega gv^-}. \end{gather*} Then, using the sign conditions in (B4) we obtain \begin{gather*} {\int_\Omega |\nabla u^-|^{p} \leq a\int_\Omega m(x)|u^-|^{p} - b\int_\Omega m_1(x)h(u,-v^-)u^{-}}\\ {\int_\Omega |\nabla v^-|^{q} \leq d\int_\Omega n(x)|v^-|^{q} - c\int_\Omega n_1(x)k(-u^-,v)v^-}. \end{gather*} Recalling the conditions in (B4), we derive that \begin{gather*} h(u,-v^-)u^{-} = -\Gamma^{\alpha + \beta +2-p}(u^-)^{\alpha+1} (v^-)^{\beta+1} ,\\ k(-u,v)v^- = -\Gamma^{\alpha + \beta +2-q}(u^-)^{\alpha+1} (v^-)^{\beta+1} \end{gather*} and hence \begin{gather*} {\int_\Omega |\nabla u^-|^{p} \leq a\int_\Omega m|u^-|^{p} + b\Gamma^{\alpha +\beta +2 - p} \int_\Omega m_1(x) (u^{-})^{\alpha + 1}(v^-)^{\beta + 1}}\\ {\int_\Omega |\nabla v^-|^{q} \leq d\int_\Omega n|v^-|^{q} + c\Gamma^{\alpha + \beta+ 2-q} \int_\Omega n_1(x) (u^-)^{\alpha + 1}(v^{-})^{\beta + 1}}. \end{gather*} Combining the variational characterization of $\lambda_1(m,p)$ and $\lambda_1(n,q)$ with the H\"{o}lder inequality and assumption (B3), we have \begin{align*} &(\lambda_1(m,p) - a)\int_\Omega m(x)|u^-|^{p} \\ &\leq b\Gamma^{\alpha + \beta+2- p} \Big(\int_\Omega m(x)|u^-|^p\Big) ^{(\alpha + 1)/q} \Big(\int_\Omega (n(x)|v^-|^q)\Big)^{(\beta + 1)/p}, \end{align*} \begin{align*} &(\lambda_1(n,q) - d)\int_\Omega n(x)|v^-|^{q}\\ &\leq c\Gamma^{\alpha + \beta +2 -q} \Big(\int_\Omega m(x)|u^-|^p\Big)^{(\alpha + 1)/q} \Big(\int_\Omega (n(x)|v^-|^q) \Big)^{(\beta + 1)/p}, \end{align*} which implies \begin{equation} \label{a3} \begin{gathered} \begin{aligned} &\Big(\int_\Omega m(x)|u^-|^p\Big)^{(\alpha + 1)/p} \Big[(\lambda_1(m,p) - a)\Big(\int_\Omega m(x)|u^-|^p\Big)^{(\beta + 1)/q}\\ &-b\Gamma^{\alpha +\beta +2 -p} \Big(\int_\Omega n(x)|v^-|^q\Big)^{(\beta + 1)/q} \Big]\leq 0, \end{aligned} \\ \begin{aligned} &\Big(\int_\Omega n(x)|v^-|^q\Big)^{(\beta + 1)/q} \Big[(\lambda_1(n,q) - d)\Big(\int_\Omega n(x)|v^-|^q\Big)^{(\alpha + 1)/p} \\ &-c\Gamma^{\alpha + \beta +2 -q} \Big(\int_\Omega m(x)|u^-|^p\Big)^{(\alpha + 1)/p} \Big]\leq 0. \end{aligned} \end{gathered} \end{equation} Let us show that $u^- = v^- = 0$. $\bullet$ If $\int_{\Omega}m(x)|u^-|^p = 0$ or $\int_{\Omega}n(x)|v^-|^q = 0$ then, using the fact that $m > 0 $, $n > 0 $, and \eqref{a3}, we obtain $u^- = v^- = 0$, which implies that the Maximum Principle holds. $\bullet$ If, $\int_{\Omega}m(x)|u^-|^p \not= 0$ and $\int_{\Omega}n(x)|v^-|^p \not= 0$, then we have \begin{gather*} %(3.5) {(\lambda_1(m,p) - a)\Big(\int_\Omega m(x)|u^-|^p \Big)^{(\beta + 1)/q} \leq b\Gamma^{\alpha + \beta +2 -p} \left( \int_\Omega n(x)|v^-|^q\right)^{(\beta + 1)/q} }\\ { (\lambda_1(n,q) - d)\Big(\int_\Omega n(x)|v^-|^q\Big)^{(\alpha + 1)/p} \leq c\Gamma^{\alpha + \beta +2 -q} \Big( \int_\Omega m(x)|u^-|^p\Big)^{(\alpha + 1)/p}}, \end{gather*} which implies \begin{gather*}% \label{a4} \begin{split} &\Big(\lambda_1(m,p) - a\Big)^{(\alpha + 1)/ p} \Big(\int_\Omega m(x)|u^-|^p\Big) ^{\frac{\alpha + 1}{p}\frac{\beta + 1}{q}}\\ &\leq b^{\alpha + 1\over p} \Gamma^{(\alpha + \beta +2 -p) \frac{\alpha +1}{p}} \Big(\int_\Omega n(x)|v^-|^q\Big) ^{\frac{\alpha + 1} {p}\frac{\beta + 1}{q}}, \end{split}\\ \begin{split} &(\lambda_1(n,q) - d)^{(\beta + 1)/q} \Big(\int_\Omega n(x)|v^-|^q\Big) ^{\frac{\beta + 1 }{q}\frac{\alpha + 1}{p}} \\ &\leq c^{(\beta + 1)/q}\Gamma^{(\alpha + \beta +2 -q) {\frac{\beta +1}{q}}}\Big(\int_\Omega m(x)|u^-|^p\Big)^ {\frac{\beta + 1}{q}\frac{\alpha + 1}{p}}. \end{split} \end{gather*} Multiplying the two inequalities above and using the fact that \begin{equation} \label{f1} \begin{split} &(\alpha + \beta + 2-p)\frac{\alpha + 1}{p} + (\alpha + \beta + 2-q)\frac{\beta +1}{q}\\ &=(\alpha + \beta + 2)(\frac{\alpha+1}{p} +\frac{\beta + 1}{q})-(\alpha +1)- (\beta + 1) = 0 \end{split} \end{equation} one has \begin{align*} &(\lambda_1(m,p) - a)^{(\alpha + 1)/ p} (\lambda_1(n,q) - d)^{(\beta + 1)/q}\\ &\times \Big[\Big(\int_\Omega m(x)|u^-|^p\Big) \Big(\int_\Omega n(x)|v^-|^q\Big)\Big] ^{\frac{\alpha+1}{p}\frac{\beta + 1}{q}}\\ &\leq b^{\alpha + 1\over p}c^{(\beta + 1)/q} \Big[\Big(\int_\Omega m(x)|u^- |^p\Big)\Big(\int_\Omega n|v^-|^q\Big)\Big] ^{\frac{\alpha + 1}{p}\frac{\beta + 1}{q}} \end{align*} and then \begin{align*} &\big[(\lambda_1(m,p) - a)^{(\alpha + 1)/ p} (\lambda_1(n,q) - d)^{(\beta + 1)/q} - b^{\alpha + 1\over p}c^{(\beta + 1)/q}\big]\\ &\times \Big[\Big(\int_\Omega m(x)|u^- |^p\Big) \Big(\int_\Omega n(x)|v^-|^q\Big) ] ^{\frac{\alpha + 1}{p}\frac{\beta + 1}{q}} \leq 0 \end{align*} Since (C1)--(C3) are satisfied, the inequality above is not possible. Consequently $u^- = v^- = 0$ and the Maximum Principle holds. \end{proof} When $p = q$ and $m = n $, the number $\theta$ is equal to $1$ and as a consequence of Theorem \ref{thm1}, we have the following result. \begin{corollary} \label{coro2}. Consider the cooperative system \eqref{eS} with $p=q > 1$ and $m=n$. Then the Maximum Principle holds if and only if {\rm (C1)--(C3)} are satisfied. \end{corollary} \begin{remark} \label{rem3} \rm Our result is reduced to the one in \cite{bouch} when $h(s,t) = |s|^{\alpha}|t|^{\beta}t$, $ k(s,t) = |s|^{\alpha}s|t|^{\beta}$ and $m= n= 1$. When $p=q$ and $\alpha = \beta = p-2$, we obtain the result in \cite{fleck2}. \end{remark} \section{Existence of Solutions} We prove in this section that, under some conditions, system \eqref{eS} admits at least one solution. \begin{theorem} \label{thm4} Assume {\rm (B1), (B2), (C1), (C2), (C3)} are satisfied. Then for $f\in L^{p'}(\Omega)$ and $g\in L^{q'}(\Omega)$, system \eqref{eS} admits at least one solution in $W_0^{1, p}(\Omega)\times W_0^{1, q}(\Omega)$. \end{theorem} The proof will be given in several steps. It borrows some ideas from \cite{bouch, lea}, and requires the Lemmas state below. We choose $r>0$ such that $a+r> 0$ and $d+r > 0$. Hence \eqref{eS} reads as follows: \begin{equation} \label{eSr} \begin{gathered} -\Delta_p u + rm(x)|u|^{p-2}u = (a+r)m(x)|u|^{p-2}u + bn_1(x)h(u,v) + f \quad\text{in } \Omega\\ -\Delta_q v + rn(x)|v|^{p-2}v = cn_1k(u,v)+ (d+r)n(x)|v|^{p-2}v + g \quad\text{in } \Omega \\ u = v = 0 \quad \text{on } \partial{\Omega} \end{gathered} \end{equation} Following \cite{boc} and \cite{bouch}, for $0<\epsilon<1$, we introduce the system \begin{equation} \label{eSe} \begin{gathered} -\Delta_p u_\epsilon + rm(x)|u_\epsilon|^{p-2}u_\epsilon = \hat{h}(x,u_\epsilon, v_\epsilon) + f \quad\text{in } \Omega\\ -\Delta_q v_\epsilon + rn(x)|v_\epsilon|^{q-2}v_\epsilon = \hat{k}(x,u_\epsilon, v_\epsilon)+ g \quad\text{in } \Omega \\ u _\epsilon = v_\epsilon = 0 \quad \text{on } \partial{\Omega} \end{gathered} \end{equation} where \begin{gather*} \hat{h} (x,s, t) = (a+r)m(x)|s|^{p-2}s(1 +\epsilon^ {1 \over p }|s|^{p-1})^{-1} + bm_1(x)h(s,t)(1 +\epsilon|h(s,t)|)^{-1}, \\ \hat{k} (x,s, t) = (d+r)n(x)|t|^{p-2}t (1 + \epsilon^{1/q}|t|^{q-1})^{-1} + cn_1 k(s,t) (1 + \epsilon| k(s,t)|)^{-1} \end{gather*} \begin{lemma} \label{lem1} System \eqref{eSe} has a solution in $W_0^{1, p}(\Omega)\times W_0^{1, q}(\Omega)$ \end{lemma} \begin{proof} Let $\epsilon > 0$ be fixed $\bullet$ Construction of sub-solution and super-solution for system \begin{equation} \label{a5} %(I) \begin{gathered} -\Delta_p u + rm(x)|u|^{p-2}u = \hat{h}(x,u, v) + f \quad\text{in } \Omega\\ -\Delta_q v + rn(x)|v|^{p-2}v = \hat{k} (x,u, v) + g \quad\text{in } \Omega \\ u = v = 0 \quad \text{on } \partial{\Omega} \end{gathered} \end{equation} From (B3), the functions $\hat{h}$ and $\hat{k}$ are bounded; that is, there exists a positive constant $M$ such that $$ |\hat{h}(x,u , v)|< M,\quad |\hat{k}(x,u, v)|< M \quad \forall (u, v)\in W_0^{1, p}(\Omega) \times W_0^{1, q}(\Omega) $$ Let $ u^0 \in W_0^{1, p}(\Omega)$ (respectively $v^0 \in W_0^{1, q}(\Omega)$) be a solution of \begin{gather*} -\Delta_p u^0 + rm(x)|u^0|^{p-2}u^0 = M + f \\ (\text{resp. } -\Delta_p v^0 + rn(x)|v^0|^{q-2}v^0 = M + g ) \end{gather*} and $u_0 \in W_0^{1, p}(\Omega)$ (resp $v_0 \in W_0^{1, q}(\Omega)$) be a solution of equation $$ \Delta_p u_0 + rm(x)|u_0|^{p-2}u_0 = -M + f \mbox {(resp}-\Delta_p v_0 + rn(x)|v^0|^{q-2}v_0 = -M + g ) $$ The existence of $u_0 ,u^0 , v_0 ,v^0$ is proved in \cite{lion}. Moreover we have \begin{gather*} -\Delta_p u_0 + rm(x)|u_0|^{p-2}u_0 - \hat{h}(x,u_0, v) - f \leq 0 \quad \forall v\in [v_0, v^0],\\ -\Delta_p u^0 + rm(x)|u^0|^{p-2}u^0 - \hat{h}(x,u^0, v) - f \geq 0 \quad \forall v\in [v_0, v^0],\\ -\Delta_q v_0 + rn(x)|v_0|^{q-2}v_0 - \hat{k}(x,u ,v_0) - g \leq 0 \quad \forall u\in [u_0, u^0], \\ -\Delta_q v^0 + rn(x)|v^0|^{q-2}v^0 - \hat{k}(x, u ,v^0) - g \geq 0 \quad \forall u\in [u_0, u^0] \end{gather*} So $(u_0, u^0)$ and $(v_0, v^0)$ are sub-super solutions of \eqref{a5}. $\bullet$ Let $K = [u_0, u^0]\times[v_0, v^0]$ and let $T: (u, v) \mapsto (w, z)$ the operator such that \begin{equation} \label{e4.2} \begin{gathered} -\Delta_p w + rm(x)|w|^{p-2}w = \hat{h}(x,u, v) + f \quad \text{in } \Omega\\ -\Delta_q z+ rn(x)|z|^{q-2}z = \hat{k} (x,u, v) + g \quad \text{in } \Omega \\ u = v = 0 \quad \text{on } \partial{\Omega}. \end{gathered} \end{equation} $\bullet$ Let us prove that $T(K)\subset K$. If $(u,v)\in K$, then \begin{equation}\label{a10'} -(\Delta_pw-\Delta_p\xi^0)+rm(x)(|w|^{p-2}w-|\xi^0|^{p-2}\xi^0 ) =[\hat{h}(x,u,v)-M]) \end{equation} Taking $(w-\xi^0)^+$ as test function in \eqref{a10'}, we have \begin{align*} &\int_{\Omega}(|\nabla w|^{p-2}\nabla w-|\nabla \xi^0|^{p-2} \nabla \xi^0) \nabla (w-\xi^0)^+\\ &+r\int_{\Omega}m(x)(|w|^{p-2}w-|\xi^0|^{p-2}\xi^0)(w-\xi^0)^+\\ &=\int_{\Omega}[(h(x,u,v)-M)](w-\xi^0)^+\leq 0. \end{align*} Since the weight $m$ is positive, by the monotonicity of the function $s\mapsto |s|^{p-2}s$ and that of the p-Laplacian, we deduce that the last integral equal zero and then $(w-\xi^0)^+=0$; that is, $w\leq \xi^0$. Similarly we obtain $\xi_0\leq w$ by taking $(w-\xi_0)^-$ as test function in \eqref{a10'}. So we have $\xi_0\leq w\leq \xi^0$ and $\eta_0\leq z\leq \eta^0$ and the step is complete. $\bullet$ To show that $T$ is completely continuous we need the following Lemma. \begin{lemma} \label{lem2} If $(u_n, v_n)\to (u, v)$ in $L^p(\Omega)\times L^q(\Omega)$ as $n\to \infty$, then \begin{itemize} \item[(1)] $X_n = {m(x)\frac{|u_n|^{p-2}u_n}{1 + |\epsilon ^{1/p}u_n|^{p-1}}}$ converges to $X= {m(x)\frac{|u|^{p-2}u}{1 + |\epsilon ^{1/p}u|^{p-1}}}$ in $L^{p'}(\Omega)$ as $n\to \infty$. \item[(2)] $Y_n = {m_1(x)\frac{h(u_n,v_n)}{1 +\epsilon|h(u_n,v_n)|}}$ converges to $Y={m_1(x)\frac{h(u,v)}{1 +\epsilon|h(u,v)|}}$ in $L^{q'}(\Omega)$ as $n\to\infty$. \end{itemize} \end{lemma} \begin{proof} Since $u_n \to u$ in $L^p(\Omega)$, there exists a subsequence still denoted $(u_n)$ such that \begin{equation} \label{e4.4} \begin{gathered} u_n(x) \to u(x) \quad\text{a.e. on } \Omega,\\ |u_n(x)|\leq \eta(x)\quad \text{a.e. on $\Omega$ with } \eta \in L^p(\Omega) \end{gathered} \end{equation} Let $$ {X_n = m(x)\frac{|u_n|^{p-2}u_n} {1 + |\epsilon^{1/p}u_n|^{p-1}}}\,. $$ Then \begin{gather*} X_n(x)\to X(x) = m(x)\frac{|u(x)|^{p-2}u(x)} {1 + |\epsilon ^{1/p}u(x)|^{p-1}}\quad \text{a.e. on } \Omega, \\ |X_n| \leq\|m\|_{\infty}|u_n|^{p-1}\leq \|m\|_{\infty}|\eta|^{p-1} \end{gather*} in $L^{p'}(\Omega)$. Thus, from Lebesgue's dominated convergence theorem one has $$ X_n \to X = m(x)\frac{|u|^{p-2}u}{1 + |\epsilon ^{1/p}u|^{p-1}} \quad \text{ in } L^{p'}(\Omega) \quad \text{as }n\to \infty $$ So (1) is proved. Moreover, since $v_n \to v$ in $L^q(\Omega)$, there exists a subsequence still denoted $(v_n)$ such that \begin{equation} \label{e4.5} \begin{gathered} v_n(x) \to v(x) \quad \text{a.e on} \Omega, \\ |v_n(x)|\leq \zeta(x) \quad \text{a.e on $\Omega$ with} \zeta \in L^q(\Omega) \end{gathered} \end{equation} Using (B4), one has $$ |Y_n| \leq \|m_1\|_{\infty}|h(u_n,v_n)|\leq \Gamma^{\alpha +\beta + 2-p}\|m_1\|_{\infty}|\eta|^{\alpha} |\zeta|^{\beta+1} $$ in $L^{p'}(\Omega)$, since $\frac{\alpha}{p} + \frac{\beta +1}{q} = {1\over p'}$. Let $$ Y_n = m_1(x)\frac{h(u_n,v_n)}{1 + \epsilon|h(u_n,v_n)|} $$ Then $$ Y_n(x) \to Y(x) = m_1(x)\frac{h(u(x), v(x))}{1 + \epsilon|h(u(x),v(x))|} \quad \text{a.e in } \Omega $$ So, we can apply the Lebesgue's dominated convergence theorem and then we obtain $$ Y_n(x) \to Y(x) = m_1(x)\frac{h(u(x), v(x))}{1 + \epsilon|h(u(x),v(x))|} \quad \text{in } L^{p'}(\Omega). $$ as $n\to \infty$. \end{proof} \begin{remark} \label{rem7} \rm We can similarly prove that, as $n\to \infty$, \begin{gather*} n(x)|v_n|^{q-2}v_n(1+|\epsilon^{1/q}v_n|^{q - 1})^{-1}\to n(x)|v|^{q-2}v(1+|\epsilon^{1/q}v|^{q - 1})^{-1} \quad \text{in} L^{q'}(\Omega), \\ n_1(x)k(u_n,v_n)(1+\epsilon|k(u_n,v_n)|)^{-1} \to n_1(x)k(u,v)(1+\epsilon|k(u,v)|)^{-1}\quad\text{in }L^{q'} (\Omega) \end{gather*} \end{remark} $\bullet$ To complete the continuity of $T$. Let us consider a sequence $(u_n, v_n)$ such that $(u_n, v_n)\to (u, v)$ in $L^{p}(\Omega)\times L^{q}(\Omega)$ as $n\to \infty$. We will prove that $(w_n, z_n)= T(u_n, v_n)\to (w, z)= T(u,v)$. Note that $(w_n, z_n)= T(u_n, v_n)$ if only if \begin{equation} \label{b1} \begin{aligned} &(-\Delta_pw_n + rm(x)|w_n|^{p-2}w_n) - (-\Delta_pw + rm(x)|w|^{p-2}w)\\ &= \hat{h}(x,u_n, v_n)- \hat{h}(x,u, v)\\ &=(a+r)[m(x)\frac{|u_n|^{p-2}u_n}{1+|\epsilon^{1/p}u_n|^{p-1}}- m(x)\frac{|u|^{p-2}u}{1+|\epsilon^{1/p}u|^{p-1}}]\\ &\quad +b m_1(x)[ \frac{h(u_n,v_n)}{1+\epsilon|h(u_n,v_n|} -\frac{h(u,v)}{1+\epsilon|h(u,v)|}]\\ &= (a +r)(X_n - X) + b(Y_n - Y) \end{aligned} \end{equation} Multiplying by $(w_n-w)$ and integrating over $\Omega$ one has \begin{align*} &\int_\Omega (|\nabla w_n|^{p-2}\nabla w_n- |\nabla w|^{p-2} \nabla w)\nabla(w_n -w)\\ &+r\int_\Omega m(x)(|w_n|^{p-2}w_n- |w|^{p-2}w).(w_n - w)\\ &= (a+r)\int_\Omega(X_n - X)(w_n -w) + b\int_\Omega (Y_n - Y)(w_n-w)\\ &\leq (a+r)\Big(\int_\Omega |X_n - X|^{p'}\Big)^{1/p'} \Big(\int_\Omega |w_n - w|^p)^{1/p}\\ &\quad + b\Big(\int_\Omega |Y_n - Y|^{p'}\Big)^{1/p'} \Big(\int_ \Omega |w_n - w|^{p}\Big)^{1/p} \end{align*} Combining Lemma \ref{lem2} and the inequality \begin{equation} \label{a12} \|x-y\|^p\leq c[(\|x\|^{p-2}x-\|y\|^{p-2}y)(x-y)]^{s/2} [\|x\|^p+\|y\|^p]^{1-s/2}, \end{equation} where $x,y\in\mathbb{R^N}$, $c=c(p)>0$ and $s=2$ if $p\geq 2$, $s=p$ if $1