\documentclass[twoside]{article} \usepackage{amssymb, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A nonexistence result \hfil EJDE--2002/56} {EJDE--2002/56\hfil Mokthar Kirane \& Eric Nabana \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 56, pp. 1--11. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A nonexistence result for a system of quasilinear degenerate elliptic inequalities in a half-space % \thanks{ {\em Mathematics Subject Classifications:} 35D05, 35J99. \hfil\break\indent {\em Key words:} Elliptic systems, nonexistence. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted March 02, 2002. Published June 17, 2002.} } \date{} % \author{Mokthar Kirane \& Eric Nabana} \maketitle \begin{abstract} We show that a system of quasilinear degenerate elliptic inequalities does not have non-trivial solutions for a certain range of parameters in the system. The proof relies on a suitable choice of the test function in the weak formulation of the inequalities. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction} For $N\geq 2$, let $\Omega =\mathbb{R}^N_+=\{ (x',x_N): x'\in\mathbb{R}^{N-1}, x_N>0\}$ and $\partial\Omega$ its boundary. On this domain, we consider the system \begin{equation}\label{system} \begin{gathered} -\vert x\vert^{\alpha}\Delta u \geq \vert v\vert^p,\\ -\vert x\vert^{\beta}\Delta v \geq \vert u\vert^q, \end{gathered} \end{equation} which can be viewed as the elliptic part of a system of wave equations where the velocity in each equation vanishes near $x=0$. This accounts for the effect of a medium that is dense near $x=0$. \paragraph{Definition} The couple $(u,v)$ is called a solution of (\ref{system}), if \begin{gather*} u\in L^1(\partial\Omega)\cap L_{{\rm loc}}^q(\Omega,|x|^{-\beta}\,dx), \quad v\in L^1(\partial\Omega)\cap L_{{\rm loc}}^p(\Omega,|x|^{-\alpha}\,dx),\\ \partial u / \partial \nu, \; \; \partial v / \partial \nu \in L^1_{{\rm loc}}(\partial\Omega)\,. \end{gather*} and for every positive regular function $\psi$, \begin{gather*} -\int_\Omega u\Delta\psi - \int_{\partial\Omega} \frac{\partial u}{\partial\nu}\,\psi + \int_{\partial\Omega} \frac{\partial \psi}{\partial\nu}\,u \geq \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi, \\ - \int_\Omega v\Delta\psi - \int_{\partial\Omega} \frac{\partial v}{\partial\nu}\,\psi + \int_{\partial\Omega} \frac{\partial \psi}{\partial\nu}\,v \geq \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi. \end{gather*} NOTATION. We let $L_{{\rm loc}}^m(\Omega,|x|^{-\delta}\,dx)$ be the set of all functions $ f:\Omega\to \mathbb{R} $ such that for every compact set $K\subseteq\Omega $, $ \int_{K}|f|^m\,|x|^{-\delta}\,dx <\infty$. Before we present our results, let us dwell a moment on some previous interesting articles. In their celebrated article, Br\'ezis and Cabr\'e \cite{BC} considered the problem \begin{gather*} -\vert x\vert^{2}\Delta u \geq u^2,\quad x\in D, \\ u=0,\quad \hbox{on } \partial D, \end{gather*} where $D$ is a smooth bounded domain of $ \mathbb{R}^N$ containing $0$. They proved that it admits as a weak solution only the trivial solution. Moreover, they gave nonexistence results of weak positive solutions for general equations of the form \[ -\Delta u = a(x)g(u) + b(x),\quad x\in D,\] under some assumptions on $a(x)$ and $b(x)$, with $g$ a continuous function on $\mathbb{R}$, nondecreasing on $\mathbb{R}^+$, such that $\int_1^{\infty} (1/g(s)) ds<\infty$. On the other hand, Esteban and Giacomoni in \cite{EG} studied the structure of the set of solutions to the problem \begin{gather*} -\vert x\vert^{2}\Delta u = \lambda u + g(u), \quad x\in B=\{ x\in\mathbb{R}^N : |x|<1\},\\ u\geq 0 \quad \hbox{in} \quad B,\\ u=0,\quad \hbox{on }\partial B. \end{gather*} Concerning equations posed in a half-space, Chipot, Chleb\'{\i}k, Fila and Shafrir \cite{CCFS} considered the problem \begin{gather*} -\Delta u = a u^p,\quad \hbox{on } \Omega=\mathbb{R}^N_+,\\ -\frac{\partial u}{\partial x_N}=u^q,\quad \hbox{on } \partial \Omega, \end{gather*} where $a\geq 0$ and $p,q>1$. They proved the existence of positive solutions, for \[ p\geq \frac{N+2}{N-2}\quad\hbox{and}\quad q \geq \frac{N}{N-2},\] and obtained nonexistence results for $a>0$ when one of the following requirements is satisfied: \begin{enumerate} \item[(i)] $p\leq \frac{N+2}{N-2}\quad$ and $\quad q \leq \frac{N}{N-2}$ with at least one strict inequality, \item[(ii)] $p < \frac{N}{N-2}$, \item[(iii)] $q < \frac{N}{N-1}$. \end{enumerate} Concerning our results, they can be summarized as follows: In section 2, we show that (\ref{system}) cannot admit nontrivial solutions $(u,v)$ for some range of $p$ and $q$ whenever \[\int_{\{ x_N=0\}}(u+v)\,dx'> 0 \,.\] However in section 3, we treat the particular case of positive solutions to (\ref{system}) and obtain different results under conditions different from those of section 2. This is due to the methods employed. Furthermore, in each section, nonexistence results are extended to systems of $m\geq 2$ inequalities. \section{Nonexistence via Young's inequality} \begin{theorem} \label{thm2.1} Assume $p>1$, $q>1$, $\alpha\leq 2$, $\beta\leq 2$, and that \[ N\leq \min \Big( \frac{p+1-\alpha}{p-1},\frac{q+1-\beta}{q-1} \Big). \] Then, there exist no nontrivial solutions $(u,v)$ of the problem (\ref{system}) such that \begin{equation} \label{C} (u+v)\vert_{x_N=0} \in L^1(\mathbb{R}^{N-1}), \quad \int_{\{ x_N=0\}}(u+v)\,dx'> 0 . \end{equation} \end{theorem} \begin{remark} \label{rmk2.1} \rm Observe that in the usual case where $\alpha=\beta=0$, we have nonexistence for $N\geq 2$ and \[ 1 0 .\] Let $\varphi$ be a positive test function in ${\cal C}^2(\Omega)$, $\varphi$ decreasing, and $\varphi(x)=\varphi_0^{\lambda}\left(\vert x\vert/R\right)$, where $(R>0)$ and \[ \varphi_0(\xi)=\begin{cases} 1 &\mbox{if } 0\leq \xi\leq 1\\ 0 &\mbox{if }\xi\geq 2. \end{cases} \] The parameter $\lambda$ will be specified later. Let $\psi(x)=x_N\varphi(x)\geq 0$. Then \begin{gather*} \frac{\partial\psi}{\partial{x_N}}=\varphi(x)+ x_N \frac{\partial\varphi}{\partial{x_N}},\quad \nabla_{x'}\psi=x_N \nabla_{x'}\varphi,\\ \Delta\psi = \frac{\partial^2\psi}{\partial{x_N^2}} + \sum_{i=1}^{N-1}\frac{\partial^2\psi}{\partial{x_i^2}} = 2\frac{\partial\varphi}{\partial{x_N}} + x_N \frac{\partial^2\varphi}{\partial{x_N^2}} + x_N \sum_{i=1}^{N-1}\frac{\partial^2\varphi}{\partial{x_i^2}}. \end{gather*} Since $\int_{\{ x_N=0\}} ( \partial u / \partial \nu) \psi=0$ and $\partial \psi / \partial \nu =- \partial \psi / \partial x_N $, from the above definition we obtain \[ -\int_\Omega \Delta\psi u - \int_{\{ x_N=0\}}\frac{\partial \psi}{\partial x_N}u \geq \int_\Omega \vert v\vert^p\,\vert x\vert^{-\alpha}\psi. \] Since $ \partial \psi / \partial x_N (x',0)=\varphi$, we have \[ \int_\Omega \vert v\vert^p\,\vert x\vert^{-\alpha} \, \psi \leq -\int_\Omega \Delta\psi \, u - \int_{\{ x_N=0\}}\varphi \, u. \] Then it follows that \begin{equation}\label{int1} \int_\Omega \vert v\vert^p\,\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}}\varphi u \leq \int_\Omega \vert\Delta\psi\vert\vert u \vert. \end{equation} We have also \begin{equation}\label{int2} \int_\Omega \vert u\vert^q\,\vert x\vert^{-\beta}\psi + \int_{\{ x_N=0\}}\varphi v \leq \int_\Omega \vert\Delta\psi\vert\vert v\vert . \end{equation} Now, using (2.2), (2.3) and Young's inequality, we obtain \[ \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi + \int_{\{ x_N=0\}}\varphi v \leq \varepsilon \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + C(\varepsilon)\int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'} \vert x\vert^{\alpha(p'-1)}, \] with $p+p'=pp'$, and \[ \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}}\varphi u \leq \varepsilon \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi +C(\varepsilon)\int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'} \vert x\vert^{\beta(q'-1)}, \] with $q+q'=qq'$. Therefore, \begin{multline*} (1-\varepsilon)\int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi +(1-\varepsilon)\int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}} (u + v)\varphi \\ \leq C(\varepsilon)\int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'} \vert x\vert^{\alpha(p'-1)} + C(\varepsilon)\int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'} \vert x\vert^{\beta(q'-1)}. \end{multline*} Hence for $0< \varepsilon <1$, there exists $C>0$ such that \begin{multline}\label{estimation1} \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi+ \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}} (u + v)\varphi \\ \leq C\Big(\int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'}\vert x\vert^{\alpha(p'-1)} + \int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'}\vert x\vert^{\beta(q'-1)} \Big). \end{multline} At this stage, we introduce the scaled variables: \[ \eta=(\eta_1,\cdots,\eta_N)=R^{-1}x=(R^{-1}x_1,R^{-1}x_2,\cdots,R^{-1}x_N). \] We have \[ \Delta\psi =R^{-1} \Big( 2\frac{\partial\varphi_0^\lambda}{\partial{\eta_N}} +\eta_N \Delta\varphi_0^\lambda\Big) =:R^{-1}A(\eta). \] It is clear that the support of $ \partial\varphi_0^\lambda / \partial{\eta_N}$ and the support of $\Delta\varphi_0^\lambda$ are subsets of ${\cal C}:= \{ \eta\in\mathbb{R}:1\leq\vert\eta\vert\leq 2 \}$. The relation (\ref{estimation1}) is then written \begin{multline*} \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi + \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}} (u + v)\varphi\\ \leq C_1R^{N+ p'(\alpha-2)+(1-\alpha)} + C_2R^{N+ q'(\beta-2)+(1-\beta)}. \end{multline*} where for $\lambda\gg 1$, \begin{gather*} \int_{\cal C} \frac{\vert A(\eta)\vert^{p'}\vert\eta\vert^{\alpha(p'-1)}} {|\eta_N|^{p'-1}\varphi_0^{\lambda(p'-1)}(\eta)}\,d\eta \leq C_1 <\infty, \\ \int_{\cal C} \frac{\vert A(\eta)\vert^{q'}\vert\eta\vert^{\beta(q'-1)}} {|\eta_N|^{q'-1}\varphi_0^{\lambda(q'-1)}(\eta)}\,d\eta \leq C_2 <\infty. \end{gather*} Since condition (\ref{C}) implies $ \int_{\{ x_N=0\}} (u + v)\varphi \geq 0$ for $R$ large enough, it follows that \begin{equation}\label{estimation2} \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi + \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi + \int_{\{ x_N=0\}} (u + v)\varphi \leq \tilde C R^{N+\gamma_1}, \end{equation} where $\gamma_1=\max\Big((\alpha-2)p'+1-\alpha, (\beta-2)q'+1-\beta\Big)$. It is easy to see that \[ N+\gamma_1 \leq 0\quad\Longleftrightarrow\quad N\leq\min\Big(\frac{p+1-\alpha}{p-1},\frac{q+1-\beta}{q-1}\Big). \] For $N+\gamma_1 <0$, we let $R\to\infty$ in (\ref{estimation2}) to obtain \[ \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}+ \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha} = 0 \] which implies $u=v=0$. This is a contradiction. For $N+\gamma_1 =0$, we deduce from (\ref{estimation2}) that \[ \int_\Omega \vert u\vert^q\vert x\vert^{-\beta}\psi<\infty,\quad \int_\Omega \vert v\vert^p\vert x\vert^{-\alpha}\psi <\infty \] since condition (\ref{C}) implies $\int_{\{ x_N=0\}} (u + v)\varphi \geq 0$ for large $R$. It follows that \[ \lim_{R\to\infty}\int_{\{ R\leq \vert x\vert\leq 2R\}} \vert u\vert^q\vert x\vert^{-\beta}\psi=\lim_{R\to\infty}\int_{\{ R\leq \vert x\vert\leq 2R\}} \vert v\vert^p\vert x\vert^{-\alpha}\psi=0. \] Now, we use H\"older's inequality in the right-hand side of (\ref{int1}) and (\ref{int2}) and a scaling argument as in (\ref{estimation2}) to obtain \begin{equation}\label{hold1} \begin{aligned} \int_{\{ x_N=0\}} v\varphi &+\int_\Omega \vert v\vert^p \vert x\vert^{- \alpha} \, \psi \\ \leq &\Big( \int_\Omega \vert u\vert^q \vert x\vert^{-\beta}\, \psi\Big)^{1/q} \Big( \int_\Omega \vert\Delta\psi\vert^{q'}\vert x\vert^{\beta(q'-1)}\psi^{1-q'}\Big)^{1/q'}\\ \leq& \Big( \int_{\mathop{\rm supp}\Delta \psi} \vert u\vert^q \vert x\vert^{-\beta} \psi\Big)^{1/q} \Big( C_2R^{N+ q'(\beta-2)+(1-\beta)}\Big)^{1/q'}, \end{aligned} \end{equation} and \begin{equation}\label{hold2} \begin{aligned} \int_{\{ x_N=0\}}u\varphi &+ \int_\Omega \vert u\vert^q \vert x\vert^{- \beta} \, \psi \\ \leq& \Big( \int_\Omega \vert v\vert^p \vert x\vert^{- \alpha} \, \psi\Big)^{1/p} \Big( \int_\Omega \vert\Delta\psi\vert^{p'}\vert x\vert^{\alpha(p'-1)}\psi^{1-p'} \Big)^{1/p'}\\ \leq&\Big( \int_{\mathop{\rm supp}\Delta \psi} \vert v\vert^p \vert x\vert^{- \alpha} \, \psi\Big)^{1/p} \Big( C_1R^{N+ p'(\alpha-2)+(1-\alpha)}\Big)^{1/p'}. \end{aligned} \end{equation} Since ${\mathop{\rm supp}\psi}\subset\{ R \leq \vert x\vert\leq 2R\}$, then for $N+p'(\alpha-2)+(1-\alpha)=0$ or $N+ q'(\beta-2)+(1-\beta)=0$, we let $R \to \infty$ in (\ref{hold1}) and (\ref{hold2}) to obtain, as before, \[ \int_\Omega \vert u\vert^q \vert x\vert^{-\beta} +\int_\Omega \vert v\vert^p \vert x\vert^{-\alpha} \leq 0 \quad\Longrightarrow\quad u=v=0.\] This completes the proof of Theorem \ref{thm2.1}. \hfill $\Box$ Without difficulties, we can extend the results to the system of $m$ inequalities \begin{equation}\label{sg1} \begin{gathered} -\vert x\vert^{\alpha_i}\Delta u_i\geq \vert u_{i+1}\vert^{p_i},\quad x\in\Omega,\quad 1\leq i\leq m,\\ u_{m+1}=u_1. \end{gathered} \end{equation} \begin{theorem} \label{thm2.2} Let $p_i>1$. If $p_i$ and $\alpha_i$ are such that \[ 2\leq N\leq \min_{1\leq i\leq m}\Big(\frac{p_i+1-\alpha_i}{p_i-1}\Big), \] then problem (\ref{sg1}) does not admit nontrivial solutions $(u_1,u_2,\dots ,u_m)$ satisfying $\sum_{i=1}^m u_i\vert_{x_N=0} \in L^1(\mathbb{R}^{N-1}), \quad \int_{\{ x_N=0\}}\sum_{i=1}^m u_i\,dx'> 0$. \end{theorem} \section{Nonexistence of positive solution via H\"older's inequality} \begin{theorem} \label{thm3.1} Suppose $p>1$ , $q>1$, and $\alpha$, $\beta$ satisfy \[ 1< N \leq \max \Big(\frac{pq+1-\beta+(2-\alpha)q}{pq-1}, \frac{pq+1-\alpha+(2-\beta)p}{pq-1}\Big). \] Then system (\ref{system}) does not admit nontrivial positive solutions. \end{theorem} \paragraph{Proof of Theorem \ref{thm3.1}} This proof is done by contradiction. Suppose that (\ref{system}) admits a nontrivial solution $(u,v)$ such that $u\geq 0$ and $v\geq 0$. Let $\psi$ be the same test function as in the proof of Theorem \ref{thm2.1}. Then, relations (\ref{int1}) and (\ref{int2}) become, respectively, \begin{gather}\label{int3} \int_\Omega v^p\,\vert x\vert^{-\alpha}\psi \leq \int_\Omega \vert\Delta\psi\vert\, u , \\ \label{int4} \int_\Omega u^q\,\vert x\vert^{-\beta}\psi \leq \int_\Omega \vert\Delta\psi\vert\, v . \end{gather} Now, using H\" older's inequality in the right-hand side of the above inequalities, we have \begin{gather*} \int_\Omega u^q \, \vert x\vert^{-\beta}\psi \leq \Big( \int_\Omega v^p \vert x\vert^{-\alpha}\psi\Big)^{1/p} \Big( \int_\Omega \vert\Delta\psi\vert^{p'}\psi^{1 -p'}\vert x\vert^{\alpha(p'-1)}\Big)^{1/p'}, \\ \int_\Omega v^p \vert x\vert^{- \alpha} \psi \leq \Big( \int_\Omega u^q \vert x\vert^{- \beta} \psi\Big)^{1/q} \Big(\int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'}\vert x\vert^{\beta(q'-1)}\Big)^{1/q'}, \end{gather*} where $p'= p/(p-1)$ and $q'=q/(q-1)$. Therefore, \begin{multline*} \Big(\int_\Omega u^q \vert x\vert^{- \beta} \psi\Big)^{1-1/pq} \\ \leq \Big( \int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1- q'}\vert x\vert^{\beta(q'-1)}\Big)^{1/pq'} \Big( \int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'}\vert x\vert^{\alpha(p'-1)}\Big)^{1/p'}, \end{multline*} and \begin{multline*} \Big(\int_\Omega v^p \vert x\vert^{- \alpha}\psi\Big)^{1-1/pq} \\ \leq \Big( \int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'} \vert x\vert^{\alpha(p'-1)}\Big)^{1/qp'} \Big( \int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'}\vert x\vert^{\alpha(q'-1)}\Big)^{1/q'}. \end{multline*} Using the change of variable $x=R\eta$ and choosing $\lambda$ as in the proof of Theorem \ref{thm2.1}, it follows that \begin{equation}\label{estim3} \Big(\int_\Omega u^q \vert x\vert^{-\beta}\psi\Big)^{1-1/pq} \leq C_1R^{\lambda_1}, \end{equation} where \[ \lambda_1=\frac{1}{pq}\Big\{ N(pq-1)-pq-1+\beta+(2-\alpha)q\Big\},\] and \begin{equation}\label{estim4} \Big(\int_\Omega v^p \vert x\vert^{- \alpha}\psi\Big)^{1-1/pq} \leq C_2R^{\lambda_2}, \end{equation} where \[ \lambda_2=\frac{1}{pq}\Big\{ N(pq-1)-pq-1+\alpha+(2-\beta)p\Big\}. \] Note that \[ \lambda_1\leq 0 \quad\Longleftrightarrow\quad N\leq \frac{pq+1-\beta+(2-\alpha)q}{pq-1},\] and \[ \lambda_2\leq 0 \quad\Longleftrightarrow\quad N\leq \frac{pq+1-\alpha+(2-\beta)p}{pq-1}. \] For $\lambda_1<0$ and $\lambda_2<0$, letting $R\to\infty$ in (\ref{estim3}) and (\ref{estim4}), we deduce $u=0$ and $v=0$, respectively. This is a contradiction. For $\lambda_1=0$ or $\lambda_2=0$, we can use the same argument developed in the last part of the proof of Theorem \ref{thm2.1} and show that $u=0$ or $v=0$, when $R\to\infty$. Observe that, thanks to (\ref{int3}) and (\ref{int4}), when $u=0$ then $v=0$ and vice versa. \hfill$\Box$ \smallskip To obtain a generalization of Theorem \ref{thm3.1} to the case of $m$ inequalities, we first analyze a system with three inequalities: \begin{equation}\label{system3} \begin{gathered} -\vert x\vert^{\alpha}\Delta u \geq v^p,\quad x\in\Omega,\\ -\vert x\vert^{\beta}\Delta v \geq w^q,\quad x\in\Omega,\\ -\vert x\vert^{\gamma}\Delta w \geq u^r,\quad x\in\Omega. \end{gathered} \end{equation} \paragraph{Definition} The vector $(u,v,w)$ is called a solution of (\ref{system3}), if \begin{gather*} u\in L_{{\rm loc}}^1(\partial\Omega)\cap L_{{\rm loc}}^r(\Omega,|x|^{-\gamma} \,dx),\\ v\in L_{{\rm loc}}^1(\partial\Omega)\cap L_{{\rm loc}}^p(\Omega,|x|^{-\alpha} \,dx),\\ w\in L_{{\rm loc}}^1(\partial\Omega)\cap L_{{\rm loc}}^q(\Omega,|x|^{-\beta} \,dx), \end{gather*} and for any positive regular function $\psi$ we have \begin{gather*} -\int_\Omega u\Delta\psi - \int_{\partial\Omega} \frac{\partial u}{\partial\nu}\,\psi + \int_{\partial\Omega} \frac{\partial \psi}{\partial\nu}\,u \geq \int_\Omega v^p\vert x\vert^{-\alpha}\psi, \\ \int_\Omega v\Delta\psi - \int_{\partial\Omega} \frac{\partial v}{\partial\nu}\,\psi + \int_{\partial\Omega} \frac{\partial \psi}{\partial\nu}\,v \geq \int_\Omega w^q\vert x\vert^{-\beta}\psi, \\ \int_\Omega v\Delta\psi - \int_{\partial\Omega} \frac{\partial w}{\partial\nu}\,\psi + \int_{\partial\Omega} \frac{\partial \psi}{\partial\nu}\,w \geq \int_\Omega u^r\vert x\vert^{-\gamma}\psi. \end{gather*} Let $\psi$ be defined as in the proof of Theorem \ref{thm2.1}. Then, using H\"older's inequality, we have \begin{gather*} \int_\Omega v^p \vert x\vert^{-\alpha}\psi \leq \Big( \int_\Omega u^r \vert x\vert^{-\gamma} \psi\Big)^{1/r} \Big( \int_\Omega \vert\Delta\psi\vert^{r'} \psi^{1-r'}\vert x\vert^{\gamma(r'-1)}\Big)^{1/r'}, \\ \int_\Omega w^q \vert x\vert^{-\beta}\psi \leq \Big( \int_\Omega v^p\vert x\vert^{-\alpha}\psi\Big)^{1/p} \Big( \int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'}\vert x\vert^{\alpha(p'-1)}\Big)^{1/p'} \\ \int_\Omega u^r \vert x\vert^{-\gamma}\psi \leq \Big( \int_\Omega w^q \vert x\vert^{-\beta}\psi\Big)^{1/q} \Big( \int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'}\vert x\vert^{\beta(q'-1)}\Big)^{1/q'}. \end{gather*} Put \begin{gather*} I_1= \Big( \int_\Omega \vert\Delta\psi\vert^{r'} \psi^{1-r'}\vert x\vert^{\gamma(r'-1)}\Big)^{1/r'}, \quad I_2= \Big( \int_\Omega \vert\Delta\psi\vert^{p'} \psi^{1-p'}\vert x\vert^{\alpha(p'-1)}\Big)^{1/p'}\\ I_3= \Big( \int_\Omega \vert\Delta\psi\vert^{q'} \psi^{1-q'}\vert x\vert^{\beta(q'-1)}\Big)^{1/q'}. \end{gather*} Then, we have \begin{gather*} \Big( \int_\Omega v^p \vert x\vert^{-\alpha}\psi\Big)^{(pqr-1)/p} \leq I_1^{qr}\,I_2\,I_3^q,\quad \Big( \int_\Omega w^q \vert x\vert^{-\beta}\psi\Big)^{(pqr-1)/q} \leq I_1^r\,I_2^{pr}\,I_3,\\ \Big( \int_\Omega u^r \vert x\vert^{-\gamma} \psi\Big)^{(pqr-1)/q} \leq I_1\,I_2^p\,I_3^{pq}. \end{gather*} The same change of variables used in the proof of Theorem \ref{thm2.1} gives \[ I_1= R^{\lambda_1} \left( \int_{\cal C} \frac{\vert A(\eta)\vert^{r'}\vert\eta\vert^{\gamma(r'-1)}} {|\eta_N|^{r'-1}\varphi_0^{\lambda(r'-1)}(\eta)}\, d\eta \right)^{1/r'} \] where $\lambda_1 =(N+1-\gamma)/r' + \gamma-2$. For a suitable choice of $\lambda$, we have \[ \int_{\cal C} \frac{\vert A(\eta)\vert^{r'}\vert\eta\vert^{\gamma(r'-1)}} {|\eta_N|^{r'-1}\varphi_0^{\lambda(r'-1)}(\eta)}\, d\eta <\infty. \] Therefore, there exists a constant $C_1>0$ such that \[ I_1\leq C_1 R^{\lambda_1}. \] Analogously we have \begin{gather*} I_2\leq C_2 R^{\lambda_2},\quad\hbox{with}\quad \lambda_2 =\frac{N+1-\alpha}{p'} + \alpha-2, \\ I_3\leq C_3 R^{\lambda_3},\quad\hbox{with}\quad \lambda_3 =\frac{N+1-\beta}{r'} + \beta-2. \end{gather*} It follows that \begin{gather*} \Big( \int_\Omega v^p \vert x\vert^{- \alpha}\psi\Big)^{(pqr-1)/p} \leq \tilde C_1 R^{\lambda_1qr+\lambda_2+\lambda_3q} =: \tilde C_1 R^{\sigma_1}, \\ \Big( \int_\Omega w^q \vert x\vert^{- \beta}\psi\Big)^{(pqr-1)/q} \leq \tilde C_2 R^{\lambda_1r+\lambda_2pr+\lambda_3} =: \tilde C_2 R^{\sigma_2}, \\ \Big( \int_\Omega u^r\vert x\vert^{- \gamma}\psi\Big)^{(pqr-1)/q} \leq \tilde C_3 R^{\lambda_1+\lambda_2p+\lambda_3pq} =: \tilde C_3 R^{\sigma_3}. \end{gather*} Note that \begin{gather*} \sigma_1 \leq 0 \quad\Longleftrightarrow N\leq \frac{pqr+(2-\gamma)pq+(2-\beta)p+1-\alpha}{pqr-1} = 1 + X_1, \\ \sigma_2\leq 0 \quad\Longleftrightarrow N\leq \frac{pqr+(2-\alpha)qr+(2-\gamma)q+1-\beta}{pqr-1} = 1 + X_2, \\ \sigma_3\leq 0 \quad\Longleftrightarrow N\leq \frac{pqr+(2-\beta)pr+(2-\alpha)r+1-\gamma}{pqr-1} = 1 + X_3, \end{gather*} where \begin{gather*} X_1=\frac{(2-\gamma)pq+(2-\beta)p+(2-\alpha)}{pqr-1}, \\ X_2=\frac{(2-\alpha)qr+(2-\gamma)q+(2-\beta)}{pqr-1}, \\ X_3=\frac{(2-\beta)pr+(2-\alpha)r+(2-\gamma)}{pqr-1} \end{gather*} are solutions of the linear system \begin{equation}\label{matrix} \begin{pmatrix} 1&-p&0 \cr 0&1&-q \cr -r&0&1\cr \end{pmatrix} \begin{pmatrix} X_1\cr X_2\cr X_3\cr \end{pmatrix} = \begin{pmatrix} \alpha-2\cr \beta-2\cr \gamma-2\cr \end{pmatrix}. \end{equation} We have the following nonexistence result. \begin{theorem} \label{thm3.2} Let $(X_1,X_2,X_3)^T$ be the solution of (\ref{matrix}). Then, if $N\leq X_1+1$, or $N\leq X_2+1$, or $N\leq X_3+1$, system (\ref{system3}) cannot admit nontrivial weak solutions $(u,v,w)$ such that $u\geq 0$, $v\geq 0$ and $w\geq 0$. \end{theorem} Now, we are able to announce the nonexistence result of positive solutions for the system (\ref{sg1}). \begin{theorem} \label{thm3.3} Suppose $p_i>1$ for $1\leq i\leq m$. Let $(X_1,X_2,\dots,X_m)^T$ be the solution of the linear system \[ \begin{pmatrix} 1&-p_1&0& &0&0 \cr 0&1& -p_2& &0&0 \cr \vdots&0&\ddots& &\ddots&0 \cr 0&0&0& &1&-p_{m-1} \cr -p_m&0&0& &0&1 \end{pmatrix} \begin{pmatrix} X_1\cr X_2\cr \vdots\cr X_{m-1}\cr X_m\cr \end{pmatrix} = \begin{pmatrix} \alpha_1-2\cr \alpha_2-2\cr \vdots\cr \alpha_{m-1}-2\cr \alpha_m-2\cr \end{pmatrix}. \] Then, if $N\leq 1+\max(X_1,X_2,...,X_m) $, system (\ref{sg1}) cannot admit a nontrivial positive solution. \end{theorem} \vskip 20pt \begin{thebibliography}{99} \bibitem{BC} Br\'ezis H. and X. 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(4) 29 (2000), no. 2, 393--420. \end{thebibliography} \noindent\textsc{Mokthar Kirane} \\ Laboratoire de Math\'ematiques, Universit\'e de la Rochelle, \\ Av. M. Cr\'epeau, \\ 17042 La Rochelle Cedex, France \\ e-mail: mokhtar.kirane@univ-lr.fr \smallskip \noindent\textsc{Eric Nabana} \\ LAMFA, FRE 2270, Universit\'e de Picardie, \\ Facult\'e de Math\'ematiques et d'Informatique,\\ 33, rue Saint-Leu, 80039 Amiens Cedex 01, France \\ e-mail: nabana@u-picardie.fr \end{document}