\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2006 International Conference in Honor of Jacqueline Fleckinger. \newline \emph{Electronic Journal of Differential Equations}, Conference 16, 2007, pp. 29--34. \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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{29} \title[\hfilneg EJDE/Conf/16 \hfil On positive solutions] {On positive solutions for a class of strongly coupled p-Laplacian systems} \author[J. Ali, R. Shivaji \hfil EJDE/Conf/16 \hfilneg] {Jaffar Ali, R. Shivaji} \address{ Department of Mathematics, Mississippi State University, Mississippi State, MS 39759, USA} \email[Jaffar Ali]{js415@ra.msstate.edu} \email[R. Shivaji]{shivaji@ra.msstate.edu} \thanks{Published May 15, 2007.} \subjclass[2000]{35J55, 35J70} \keywords{Positive solutions; p-Laplacian systems; semipositone problems} \dedicatory{Dedicated to Jacqueline Fleckinger on the occasion of\\ an international conference in her honor} \begin{abstract} Consider the system \begin{gather*} -\Delta_pu =\lambda f(u,v)\quad\mbox{in }\Omega\\ -\Delta_qv =\lambda g(u,v)\quad\mbox{in }\Omega\\ u=0=v \quad \mbox{on }\partial\Omega \end{gather*} where $\Delta_sz=\mathop{\rm div}(|\nabla z|^{s-2}\nabla z)$, $s>1$, $\lambda$ is a non-negative parameter, and $\Omega$ is a bounded domain in $\mathbb{R}$ with smooth boundary $\partial\Omega$. We discuss the existence of a large positive solution for $\lambda$ large when $$\lim_{x\to\infty}\frac{f(x,M[g(x,x)]^{1/q-1})}{x^{p-1}}=0$$ for every $M>0$, and $\lim_{x\to\infty} g(x,x)/x^{q-1}=0$. In particular, we do not assume any sign conditions on $f(0,0)$ or $g(0,0)$. We also discuss a multiplicity results when $f(0,0)=0=g(0,0)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\norm}[1]{\|#1\|_\infty} \newcommand{\abs}[1]{|#1|} \section{Introduction} Consider the boundary-value problem $$\label{eq1:prob} \begin{gathered} -\Delta_pu=\lambda f(u,v)\quad\text{in }\Omega\\ -\Delta_qv=\lambda g(u,v)\quad\text{in }\Omega\\ u=0=v\quad\text{on }\partial\Omega \end{gathered}$$ where $\Delta_sz=\mathop{\rm div}(|\nabla z|^{s-2}\nabla z)$, $s>1, \lambda$ is a non-negative parameter, and $\Omega$ is a bounded domain in $\mathbb{R}$ with smooth boundary $\partial\Omega$. We are interested in the study of positive solutions to \eqref{eq1:prob} when no conditions on $f(0,0),g(0,0)$ are assumed, in particular, they could be negative (semipositone systems). Semipositive problems are mathematically challenging area in the study of positive solutions (see \cite{BCN} and \cite{LP}). For a review on semipositone problems, see \cite{CMS}. In this paper we make the following assumptions: \begin{itemize} \item[(H1)] $f,g\in C^1((0,\infty)\times(0,\infty))\cap C([0,\infty)\times[0,\infty))$ be monotone functions such that $f_u,f_v,g_u,g_v\geq0$ and $\lim_{u,v\to\infty}f(u,v)= \lim_{u,v\to\infty\\}g(u,v)= \infty$. \item[(H2)] $\displaystyle \lim_{x\to\infty}\frac{f(x,M[g(x,x)]^{1/q-1})}{x^{p-1}}=0$ for every $M>0$. \item[(H3)] $\displaystyle \lim_{x\to\infty}\frac{g(x,x)}{x^{q-1}}=0$. \end{itemize} We establish the following existence and multiplicity results: \begin{theorem} \label{thm:a} Let {\rm (H1)--(H3)} hold. Then there exists a positive number $\lambda^*$ such that \eqref{eq1:prob} has a large positive solution $(u,v)$ for $\lambda>\lambda^*$. \end{theorem} \begin{theorem} \label{thm:b} Let {\rm (H1)--(H3)} hold. Further let $F(s)=f(s,cs)$ and $G(s)=g(\tilde{c}s,s)$ for any $c,\tilde{c}>0$ and assume that $f$ and $g$ be sufficiently smooth functions in the neighborhood of zero with $F(0)=G(0)=0$, $F^{(k)}(0)=0=G^{(l)}(0)$ for $k=1,2,\dots [p-1]$, $l=1,2,\dots [q-1]$ where $[s]$ denotes the integer part of $s$. Then \eqref{eq1:prob} has at least two positive solutions provided $\lambda$ is large. \end{theorem} This paper extends the recent work in \cite{AS}, where the authors study such systems with weaker coupling, namely systems of the form, $$\label{eq1:modelprob} \begin{gathered} -\Delta_pu=\lambda_1\alpha(v)+\mu_1 \delta(u)\quad\text{in }\Omega\\ -\Delta_qv=\lambda_2\beta(u)+\mu_2 \gamma(v)\quad\text{in }\Omega\\ u=0=v\quad\text{on }\partial\Omega \end{gathered}$$ where $\lambda_1,\lambda_2,\mu_1$ and $\mu_2$ are non-negative parameters, with the following conditions: \begin{itemize} \item[(C1)] $\alpha,\beta,\delta,\gamma\in C^1(0,\infty)\cap C[0,\infty)$ be monotone functions such that $$\lim_{x\to\infty}\alpha(x)= \lim_{x\to\infty}\beta(x)= \lim_{x\to\infty}\delta(x)= \lim_{x\to\infty}\gamma(x)=\infty.$$ \item[(C2)] $\displaystyle \lim_{x\to\infty}\frac{\alpha(M[\beta(x)]^{1/q-1})}{x^{p-1}}=0$ for every $M>0$. \item[(C3)] $\displaystyle \lim_{x\to\infty}\frac{\delta(x)}{x^{p-1}}= \lim_{x\to\infty}\frac{\gamma(x)}{x^{q-1}}=0$. \end{itemize} In \cite{AS}, authors establish an existence result for the system \eqref{eq1:modelprob} when $\lambda_1+\mu_1$ and $\lambda_2+\mu_2$ are large. In addition, for the case when $f(0)=h(0)=g(0)=\gamma(0)=0$, authors discuss a multiplicity result for $\lambda_1+\mu_1$ and $\lambda_2+\mu_2$ large. Here we extend this study to classes of systems with much stronger coupling. Our approach is based on the method of sub-and supersolutions (see e.g. \cite{DH}). In Section 2, we will prove Theorem \ref{thm:a}, in Section 3, we will prove Theorem \ref{thm:b} and in Section 4, we discuss some examples with strong coupling. \section{Proof of Theorem \ref{thm:a}} We extend $f(u,v)$ and $g(u,v)$ for all $(u,v)\in \mathbb{R}^2$ smoothly such that there exists a constant $k_0>0$ such that $f(u,v),g(u,v)\geq-k_0$ for all $(u,v)\in \mathbb{R}^2$. We shall establish Theorem \ref{thm:a} by constructing a positive weak subsolution $(\psi_1,\psi_2)\in W^{1,p}(\Omega)\cap C(\overline{\Omega})\times W^{1,q}(\Omega)\cap C(\overline{\Omega})$ and a supersolution $(z_1,z_2)\in W^{1,p}(\Omega)\cap C(\overline{\Omega})\times W^{1,q}(\Omega)\cap C(\overline{\Omega})$ of \eqref{eq1:prob} such that $\psi_i\leq z_i$ for $i=1,2$. That is, $\psi_i,z_i$ satisfies $(\psi_1,\psi_2)=(0,0)=(z_1,z_2)$ on $\partial\Omega$, \begin{gather*} \int_\Omega\abs{\nabla\psi_1}^{p-2}\nabla\psi_1\cdot\nabla \xi \,dx \leq\lambda\int_\Omega f(\psi_1,\psi_2)\xi \,dx,\\ \int_\Omega\abs{\nabla\psi_2}^{p-2}\nabla\psi_2\cdot\nabla \xi \,dx \leq\lambda\int_\Omega g(\psi_1,\psi_2)\xi \,dx,\\ \int_\Omega\abs{\nabla z_1}^{p-2}\nabla z_1\cdot\nabla \xi \,dx \geq\lambda\int_\Omega f(z_1,z_2)\xi \,dx,\\ \int_\Omega\abs{\nabla z_2}^{p-2}\nabla z_2\cdot\nabla \xi \,dx \geq\lambda\int_\Omega g(z_1,z_2)\xi \,dx \end{gather*} for all $\xi \in W:=\big\{\eta\in C_0^\infty(\Omega):\eta\geq0 \text{ in } \Omega\big\}$. Let $\lambda_1^{(r)}$ the first eigenvalue of $-\Delta_r$ with Dirichlet boundary conditions and $\phi_r$ the corresponding eigenfunction with $\phi_r >0;\Omega$ and $\norm{\phi_r}=1$ for $r=p,q$. Let $m,\delta>0$ be such that $\abs{\nabla\phi_r}^r-\lambda_1^{(r)}\phi_r^r\geq m$ on $\overline{\Omega}_\delta=\{x\in\Omega|d(x,\partial\Omega)\leq\delta\}$ for $r=p,q$. (This is possible since $\abs{\nabla\phi_r}\neq0$ on $\partial\Omega$ while $\phi_r=0$ on $\partial\Omega$ for $r=p,q$). We shall verify that $$(\psi_1,\psi_2):=\Big(\big[\frac{\lambda k_0}m\big]^{1/p-1}\big(\frac{p-1}p\big)\phi_p^{p/p-1}, \big[\frac{\lambda k_0}m\big]^{1/q-1}\big(\frac{q-1}q\big)\phi_q^{q/q-1}\Big),$$ is a subsolution of \eqref{eq1:prob} for $\lambda$ large. Let $\xi\in W$. Then \begin{align*} \int_\Omega\abs{\nabla\psi_1}^{p-2}\nabla\psi_1\cdot\nabla \xi\,dx &=\Big(\frac{\lambda k_0}m\Big)\int_\Omega\phi_p\abs{\nabla\phi_p}^{p-2} \nabla\phi_p\cdot\nabla \xi \,dx\\ &=\Big(\frac{\lambda k_0}m\Big)\Big\{\int_\Omega\abs{\nabla\phi_p}^{p-2} \nabla\phi_p\cdot\nabla(\phi_p \xi )\,dx-\int_\Omega\abs{\nabla\phi_p}^p\xi \,dx\Big\}\\ &=\Big(\frac{\lambda k_0}m\Big)\Big\{\int_\Omega[\lambda_1^{(p)}\phi_p^p-\abs{\nabla\phi_p}^p]\xi \,dx\Big\}. \end{align*} Similarly $\int_\Omega\abs{\nabla\psi_2}^{q-2}\nabla\psi_2\cdot\nabla \xi \,dx= \Big(\frac{\lambda k_0}m\Big)\Big\{\int_\Omega[\lambda_1^{(q)}\phi_q^q-\abs{\nabla\phi_q}^q]\xi \,dx\Big\}.$ Now on $\overline{\Omega}_\delta$ we have $\abs{\nabla\phi_r}^r-\lambda_1^{(s)}\phi_r^r\geq m$ for $r=p,q$. Which implies that \begin{gather*} \frac{k_0}m\Big(\lambda_1^{(p)}\phi_p^p-\abs{\nabla\phi_p}^p\Big) -f(\psi_1,\psi_2)\leq0 , \\ \frac{k_0}m\Big(\lambda_1^{(q)}\phi_q^q-\abs{\nabla\phi_q}^q\Big) -g(\psi_1,\psi_2)\leq0. \end{gather*} Next on $\Omega-\overline{\Omega}_\delta$ we have $\phi_p\geq\mu$, $\phi_q\geq\mu$ for some $\mu>0$, and therefore for $\lambda$ large \begin{gather*} f(\psi_1,\psi_2)\geq \frac{k_0}m\lambda_1^{(p)}\geq \frac{k_0}m\lambda_1^{(p)}\phi_p^p-\abs{\nabla\phi_p}^p,\\ g(\psi_1,\psi_2)\geq \frac{k_0}m\lambda_1^{(q)}\geq \frac{k_0}m\lambda_1^{(q)}\phi_q^q-\abs{\nabla\phi_q}^q. \end{gather*} Hence \begin{gather*} \int_\Omega\abs{\nabla\psi_1}^{p-2}\nabla\psi_1\cdot\nabla \xi \,dx\leq\lambda\int_\Omega f(\psi_1,\psi_2)\xi \,dx,\\ \int_\Omega\abs{\nabla\psi_2}^{q-2}\nabla\psi_2\cdot\nabla \xi \,dx\leq\lambda\int_\Omega g(\psi_1,\psi_2)\xi \,dx; \end{gather*} i.e., $(\psi_1,\psi_2)$ is a subsolution of \eqref{eq1:prob} for $\lambda$ large. Next let $e_r$ be the solution of $-\Delta_re_r=1$ in $\Omega$, $e_r=0$ on $\partial\Omega$ for $r=p,q$. Let $(z_1,z_2):=\Big(\frac c{\mu_p}\lambda^{1/p-1}e_p,[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]^{1/q-1}\lambda^{1/q-1}e_q\Big)$ where $\mu_r=\norm{e_r}$; $r=p,q$. Then \begin{align*} \int_\Omega\abs{\nabla z_1}^{p-2}\nabla z_1\cdot\nabla \xi \,dx &=\lambda \big(\frac c{\mu_p}\big)^{p-1}\int_\Omega\abs{\nabla e_p}^{p-2}\nabla e_p\cdot \nabla \xi \,dx\\ &=\frac1{(\mu_p)^{p-1}}(c\lambda^{1/p-1})^{p-1}\int_\Omega \xi \,dx. \end{align*} By (H2) we can choose $c$ large enough so that \begin{align*} &\frac1{(\mu_p)^{p-1}}(c\lambda^{1/p-1})^{p-1}\int_\Omega \xi \,dx \\ &\geq\lambda\int_\Omega f(c\lambda^{1/p-1},[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]^{1/q-1}\lambda^{1/q-1}\mu_q)\xi \,dx\\ &\geq\lambda\int_\Omega f(c\lambda^{1/p-1}\frac{e_p}{\mu_p},[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]^{1/q-1}\lambda^{1/q-1}e_q)\xi \,dx\\ &=\lambda\int_\Omega f(z_1,z_2)\xi \,dx. \end{align*} Next \begin{align*} \int_\Omega\abs{\nabla z_2}^{q-2}\nabla z_2\cdot\nabla \xi \,dx&=\lambda[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]\int_\Omega\abs{\nabla e_q}^{q-2}\nabla e_q\cdot\nabla\xi \,dx\\ &=\lambda[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]\int_\Omega\xi \,dx \end{align*} By (H3) choose $c$ large so that $\dfrac1{\lambda^{1/q-1}}\mu_q\geq\dfrac{[g(c\lambda^{1/p-1}, c\lambda^{1/p-1})]^{1/q-1}}{c\lambda^{1/p-1}}$, then \begin{align*} &\lambda[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]\int_\Omega\xi \,dx \\ &\geq\lambda\int_\Omega g\big(c\lambda^{1/p-1},[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]^{1/q-1}\lambda^{1/q-1}\mu_q\big)\xi \,dx\\ &\geq\lambda\int_\Omega g\big(c\lambda^{1/p-1}\frac{e_p}{\mu_p},[g(c\lambda^{1/p-1},c\lambda^{1/p-1})]^{1/q-1}\lambda^{1/q-1}e_q\big)\xi \,dx\\ &=\lambda\int_\Omega g(z_1,z_2)\xi \,dx; \end{align*} i.e., $(z_1,z_2)$ is a supersolution of \eqref{eq1:prob} with $z_i\geq\psi_i$ for $c$ large, $i=1,2$. (Note $\abs{\nabla e_r}\neq0;\partial\Omega$ for $r=p,q$). Thus, there exists a solution $(u,v)$ of \eqref{eq1:prob} with $\psi_1\leq u\leq z_1,\psi_2\leq v\leq z_2$. This completes the proof of Theorem \ref{thm:a}. \section{Proof of Theorem \ref{thm:b}} To prove Theorem \ref{thm:b}, we will construct a subsolution $(\psi_1,\psi_2)$, a strict supersolution $(\zeta_1,\zeta_2)$, a strict subsolution $(w_1, w_2)$, and a supersolution $(z_1, z_2)$ for \eqref{eq1:prob} such that $(\psi_1,\psi_2)\leq(\zeta_1, \zeta_2)\leq(z_1, z_2)$, $(\psi_1,\psi_2)\leq(w_1, w_2)\leq(z_1, z_2)$, and $(w_1, w_2) \nleq(\zeta_1, \zeta_2)$. Then \eqref{eq1:prob} has at least three distinct solutions $(u_i,v_i),~i=1,2,3$, such that $(u_1,v_1)\in[(\psi_1,\psi_2),(\zeta_1, \zeta_2)],(u_2,v_2)\in[(w_1,w_2),(z_1, z_2)]$, and $$(u_3,v_3)\in[(\psi_1,\psi_2),(z_1, z_2)] \setminus \big( [(\psi_1,\psi_2),(\zeta_1, \zeta_2)] \cup[(w_1,w_2),(z_1,z_2)]\big).$$ We first note that $(\psi_1,\psi_2)=(0,0)$ is a solution (hence a subsolution). As in Section 2, we can always construct a large supersolution $(z_1,z_2)$. We next consider $$\label{eq2:prob} \begin{gathered} -\Delta_pw_1=\lambda\tilde{f}(w_1,w_2)\quad\text{in }\Omega\\ -\Delta_qw_2=\lambda\tilde{g}(w_1,w_2)\quad\text{in }\Omega\\ w_1=0=w_2\quad\text{on }\partial\Omega \end{gathered}$$ where $\tilde{f}(u,v)=f(u,v)-1$ and $\tilde{g}(u,v)=g(u,v)-1$. Then by Theorem \ref{thm:a}, \eqref{eq2:prob} has a positive solution $(w_1,w_2)$ when $\lambda$ is large. Clearly this $(w_1,w_2)$ is a strict subsolution of \eqref{eq1:prob}. Finally we construct the strict supersolution $(\zeta_1, \zeta_2)$. To do so, we let $\phi_p,\phi_q$ as described in Section 2. We note that there exists positive constants $c_1$ and $c_2$ such that $$\label{eq:1} \phi_p\leq c_1\phi_q\quad\text{and}\quad \phi_q\leq c_2\phi_p.$$ Let $(\zeta_1, \zeta_2)=(\epsilon\phi_p,\epsilon\phi_q)$ where $\epsilon>0$. Let $H_p(s):=\lambda_1^{(p)}s^{p-1}-\lambda f(s,c_2s)$ and $H_q(s):= \lambda_1^{(q)}s^{q-1}-\lambda g(c_1s,s)$. Observe that $H_p(0)=H_q(0)=0$, $H_p^{(k)}(0)=0=H_q^{(l)}(0)$ for $k=1,2,\dots [p-2]$ and $l=1,2,\dots [q-2]$. $H_p^{(p-1)}(0)>0$ and $H_q^{(q-1)}(0)>0$ if $p,q$ are integers, while $\lim_{r\to0}H^{([p])}(r)=+\infty=\lim_{r\to0}H^{([q])}(r)$ if $p,q$ are not integers. Thus there exists $\theta$ such that $H_p(s)>0$ and $H_q(s)>0$ for $s\in(0,\theta]$. Hence for $0<\epsilon\leq\theta$ we have %(note that $\norm{\phi_p}=1=\norm{\phi_q}$) \label{eq:2} \begin{aligned} \lambda_1^{(p)}(\zeta_1)^{p-1} =\lambda_1^{(p)}(\epsilon\phi_p)^{p-1}&>\lambda f(\epsilon\phi_p, c_2\epsilon \phi_p)\\ &\geq\lambda f(\epsilon\phi_p, \epsilon \phi_q)\\ &=\lambda f(\zeta_1,\zeta_2)\quad x\in\Omega, \end{aligned} and similarly we get \label{eq:3} \begin{aligned} \lambda_1^{(q)}(\zeta_2)^{q-1} =\lambda_1^{(q)}(\epsilon\phi_q)^{q-1}&>\lambda g(c_1\epsilon\phi_q,\epsilon\phi_q)\\ &\geq\lambda g(\epsilon\phi_p, \epsilon \phi_q)\\ &=\lambda g(\zeta_1,\zeta_2), \quad x\in\Omega. \end{aligned} Using the inequalities \eqref{eq:2} and \eqref{eq:3} we have, \begin{align*} \int_\Omega\abs{\nabla \zeta_1}^{p-2}\nabla \zeta_1\cdot\nabla \xi \,dx &=\epsilon^{p-1}\int_\Omega\abs{\nabla\phi_p}^{p-2}\nabla\phi_p\cdot\nabla\xi\\ &=\int_\Omega\lambda_1^{(p)}(\epsilon\phi_p)^{p-1}\xi \,dx\\ &>\lambda\int_\Omega f(\zeta_1,\zeta_2)\xi \,dx. \end{align*} Similarly we have $$\int_\Omega\abs{\nabla \zeta_2}^{q-2}\nabla \zeta_2\cdot\nabla \xi \,dx>\lambda\int_\Omega g(\zeta_1,\zeta_2)\xi \,dx$$ Thus $(\zeta_1,\zeta_2)$ is a strict supersolution. Here we can choose $\epsilon$ small so that $(w_1,w_2)\nleq(\zeta_1,\zeta_2)$. Hence there exists solutions %\begin{gather*} $(u_1,v_1)\in[(\psi_1,\psi_2),(\zeta_1,\zeta_2)]$, $(u_2,v_2)\in[(w_1,w_2),(z_1, z_2)]$, and $(u_3,v_3)\in [(\psi_1,\psi_2),(z_1, z_2)] \setminus\big([(\psi_1,\psi_2),(\zeta_1,\zeta_2)]\cup [(w_1,w_2),(z_1, z_2)]\big).$ %\end{gather*} Since $(\psi_1,\psi_2)\equiv(0,0)$ is a solution it may turn out that $(u_1,v_1)\equiv(\psi_1,\psi_2)\equiv(0,0)$. In any case we have two positive solutions $(u_2,v_2)$ and $(u_3,v_3)$. Hence Theorem \ref{thm:b} holds. \begin{remark} \rm Note that in the construction of the supersolution $(\zeta_1,\zeta_2)$ we require the conditions at zero on $F$ and $G$ only for the constants $c=c_2$ and $\tilde{c}=c_1$. \end{remark} \section{Examples} \begin{example} \label{exa:a} \rm Consider the problem $$\label{eq1:eg1} \begin{gathered} -\Delta_p u=\lambda [v^\alpha+(uv)^\beta-1]\quad\text{in }\Omega\\ -\Delta_q v=\lambda [u^\sigma+(uv)^{\gamma/2}-1]\quad\text{in }\Omega\\ u=0=v\quad\text{on }\partial\Omega \end{gathered}$$ where $\alpha,\beta,\sigma,\gamma$ are positive parameters. Then it is easy to see that \eqref{eq1:eg1} satisfies the hypotheses of Theorem \ref{thm:a} if $\max\{\sigma,\gamma\}\frac{\alpha}{q-1}1, \end{cases}\quad\mbox{and}\quad \gamma(x)=\begin{cases} x^{\mu};&x\leq1\\ \frac{\mu}{\delta}x^{\delta}+(1-\frac{\mu}{\delta});&x>1, \end{cases}$$where$\alpha,\sigma,\mu,\delta$are positive parameters. Here we assume$\alpha>p-1 ~\mbox{if $p$ is an integer,}\alpha> [p]$if$p$is not an integer,$\mu>q-1 ~\mbox{if $q$ is an integer}$and$\mu> [q]$if$q$is not an integer. Consider the problem $$\label{eq1:eg2} \begin{gathered} -\Delta u=\lambda [1+u^\beta]h(v)\quad \mbox{in }\Omega\\ -\Delta v=\lambda \gamma(u)\quad\mbox{in }\Omega\\ u=0=v\quad\text{on }\partial\Omega \end{gathered}$$ where$0\leq\beta