\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 164, pp. 1--8.\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-2011/164\hfil Existence of positive solutions] {Existence of positive solutions for semilinear elliptic systems with indefinite weight} \author[R. Chen\hfil EJDE-2011/164\hfilneg] {Ruipeng Chen} \address{Ruipeng Chen \newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{ruipengchen@126.com} \thanks{Submitted September 13, 2011. Published December 13, 2011.} \subjclass[2000]{35J45} \keywords{Semilinear elliptic systems; indefinite weight; positive solutions; \hfill\break\indent existence of solutions} \begin{abstract} This article concerns the existence of positive solutions of semilinear elliptic system \begin{gather*} -\Delta u=\lambda a(x)f(v),\quad\text{in }\Omega,\\ -\Delta v=\lambda b(x)g(u),\quad\text{in } \Omega,\\ u=0=v,\quad \text{on } \partial\Omega, \end{gather*} where $\Omega\subseteq\mathbb{R}^N\ (N\geq1)$ is a bounded domain with a smooth boundary $\partial\Omega$ and $\lambda$ is a positive parameter. $a, b:\Omega\to\mathbb{R}$ are sign-changing functions. $f, g:[0,\infty)\to\mathbb{R}$ are continuous with $f(0)>0$, $g(0)>0$. By applying Leray-Schauder fixed point theorem, we establish the existence of positive solutions for $\lambda$ sufficiently small. \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} Let $\Omega\subseteq\mathbb{R}^N\ (N\geq1)$ be a bounded domain with a smooth boundary $\partial\Omega$ and $\lambda>0$ a parameter. Let $a, b:\Omega\to\mathbb{R}$ be sign-changing functions. We are concerned with the existence of positive solutions of the semilinear elliptic system \begin{equation} \begin{gathered} -\Delta u=\lambda a(x)f(v),\quad \text{in } \Omega,\\ -\Delta v=\lambda b(x)g(u),\quad \text{in } \Omega,\\ u=0=v,\quad \text{on } \partial\Omega. \end{gathered} \label{e1.1} \end{equation} In the past few years, the existence of positive solutions of the nonlinear eigenvalue problem \begin{equation} -\Delta u=\lambda f(u)\label{e1.2} \end{equation} has been studied extensively by many authors. It is well-known that many problems in mathematical physics may lead to problem \eqref{e1.2}. See, for example, fluid dynamics \cite{a1}, combustion theory \cite{b1,f2}, nonlinear field equations \cite{b2}, wave phenomena \cite{s1}, etc. Lions \cite{l1} studied the existence of positive solutions of Dirichlet problem \begin{equation} \begin{gathered} -\Delta u=\lambda a(x)f(u),\quad \text{in } \Omega,\\ u=0,\quad \text{on } \partial\Omega \end{gathered} \label{e1.3} \end{equation} with the weight function and nonlinearity satisfy $a\geq0$, $f\geq0$, respectively. Problem \eqref{e1.3} with indefinite weight $a(\cdot)$ is more interesting, and which has been studied by Brown \cite{b3,b4}, Cac \cite{c1}, Hai \cite{h1} and the references therein. In recent years, a good amount of research is established for reaction-diffusion systems. Reaction-diffusion systems model many phenomena in Biology, Ecology, combustion theory, chemical reactors, population dynamics etc. And the elliptic system \begin{equation} \begin{gathered} -\Delta u=\lambda f(v),\quad \text{in } \Omega,\\ -\Delta v=\lambda g(u),\quad \text{in } \Omega,\\ u=0=v,\quad \text{on } \partial\Omega \end{gathered} \label{e1.4} \end{equation} has been considered as a typical example of these models. The existence of positive solutions of \eqref{e1.4} is established by de Figueiredo \cite{f1} et al, by an Orlicz space setting for $N\geq3$. Hulshof et al \cite{h3} established the existence of positive solutions for \eqref{e1.4} by variational technique for $N\geq1$. Dalmasso \cite{d1} proved the existence of positive solutions of \eqref{e1.4} by Schauder's fixed point theorem. Hai and Shivaji \cite{h2} established the existence of positive solution of \eqref{e1.4} for $\lambda$ large, by using the method of sub and supersolutions and Schauder's fixed point theorem. Recently, Tyagi \cite{t1} studied the existence of positive solutions of \eqref{e1.1} by the method of monotone iteration and Schauder's fixed point theorem. He assumed that $a, b\in L^\infty(\Omega)$ and \begin{itemize} \item[(H1)] $f, g: [0,\infty)\to[0,\infty)$ which are continuous and nondecreasing on $[0,\infty)$; \item[(H2)] There exists $\mu_1>0$ such that $$ \int_{\Omega}G(x,y)a^+(y)dy \geq(1+\mu_1)\int_{\Omega}G(x,y)a^-(y)dy,\quad \forall x\in\Omega; $$ \item[(H3)] There exists $\mu_2>0$ such that $$ \int_{\Omega}G(x,y)b^+(y)dy \geq(1+\mu_2)\int_{\Omega}G(x,y)b^-(y)dy,\quad \forall x\in\Omega, $$ where $G(x,y)$ is the Green's function of $-\Delta$ associated with Dirichlet boundary condition. \end{itemize} Here $a^+$, $b^+$ are positive parts of $a$ and $b$; while $a^-$ and $b^-$ are the negative parts. The main result of Tyagi \cite{t1} reads as follows. \begin{theorem} \label{thmA} Assume $f(0)>0$, $g(0)>0$, $f$ and $g$ both are nondecreasing, and continuous functions. Also assume {\rm (H2), (H3)}. Then there exists $\lambda^\ast>0$ depending on $f, g, a, b, \mu_i, i=1,2$ such that \eqref{e1.1} has a nonnegative solution for $0\leq\lambda\leq\lambda^\ast$. \end{theorem} Motivated by the above references, the purpose of the present article is to study the existence of positive solutions of \eqref{e1.1} by using the Leray-Schauder fixed point theorem: \begin{lemma}[\cite{d2}] \label{lem1.1} Let $X$ be a Banach space and $T:X\to X$ a completely continuous operator. Suppose that there exists a constant $M>0$, such that each solution $(x, \sigma)\in X\times[0,1]$ of $$ x=\sigma Tx,\quad \sigma\in[0,1],\; x\in X $$ satisfies $\|x\|_X\leq M$. Then $T$ has a fixed point. \end{lemma} Next, we state the main result of this article, under the assumption \begin{itemize} \item[(H1')] $f, g: [0,\infty)\to\mathbb{R}$ are continuous with $f(0)>0, g(0)>0$. \end{itemize} \begin{theorem} \label{thm1.1} Let $a, b$ be nonzero continuous functions on $\overline{\Omega}$. Assume that {\rm (H1'), (H2), (H3)} hold. Then there exists a positive number $\lambda^\ast$ such that \eqref{e1.1} has a positive solution for $0<\lambda<\lambda^\ast$. \end{theorem} \begin{remark} \label{rmk1.1}\rm Assumption (H1') implies that the nonlinearities $f$ and $g$ can change their signs, but can not be monotone; thus (H1') is much weaker than the assumption (H1) used in Tyagi \cite{t1}. We obtain a similar result as Theorem \ref{thmA} under the weaker condition (H1'). It is worth remarking that in proving the Theorem \ref{thm1.1}, we extend the results in Hai \cite{h1}. \end{remark} As a consequence of Theorem \ref{thm1.1}, we have the following result. \begin{corollary} \label{coro1.1} Assume that (H1') holds. Let $a, b$ be nonzero integrable functions on $[0,1]$. Suppose that there exist two positive constants $k_1>1$ and $k_2>1$ such that \begin{gather*} \int_0^ts^{N-1}a^+(s)ds\geq k_1\int_0^ts^{N-1}a^-(s)ds,\quad \forall t\in[0,1],\\ \int_0^ts^{N-1}b^+(s)ds\geq k_2\int_0^ts^{N-1}b^-(s)ds,\quad \forall t\in[0,1]. \end{gather*} Then there exists a positive number $\lambda^\ast$ such that the system \begin{equation} \begin{gathered} u''+\frac{N-1}{t}u'+\lambda a(t)f(v)=0,\quad 00$ be such that \begin{equation} f(x)\geq\delta f(0),\ \ g(x)\geq\delta g(0),\quad \text{for } 0\leq x\leq\varepsilon.\label{e2.2} \end{equation} In fact, it follows from (H1') that there exist two positive constants $\varepsilon_1, \varepsilon_2$ small such that $$ f(x)\geq\delta f(0),\quad 0\leq x\leq\varepsilon_1;\quad g(x)\geq\delta g(0),\quad 0\leq x\leq\varepsilon_2. $$ Choosing $\varepsilon=\min\{\varepsilon_1,\varepsilon_2\}$, then \eqref{e2.2} holds. Define \begin{equation} \widetilde{f}(t)=\max_{s\in[0,t]}f(s),\quad \widetilde{g}(t)=\max_{s\in[0,t]}g(s),\label{e2.3} \end{equation} then $\widetilde{f}$ and $\widetilde{g}$ are continuous and nondecreasing. Let \begin{equation} \widetilde{h}(t)=\max\{\widetilde{f}(t), \widetilde{g}(t)\}, \label{e2.4} \end{equation} then $\widetilde{h}$ is continuous. Suppose that $\lambda<\frac{\varepsilon}{2\|p\|_\infty\widetilde{h}(\varepsilon)}$, thus \begin{equation} \frac{\widetilde{h}(\varepsilon)}{\varepsilon} <\frac{1}{2\lambda\|p\|_\infty},\label{e2.5} \end{equation} where $\|p\|_\infty=\max\{\|p_1\|_\infty, \|p_2\|_\infty\}$. (H1'), \eqref{e2.3} and \eqref{e2.4} imply that $\widetilde{h}(0)>0$, and therefore \begin{equation} \lim_{t\to0+}\frac{\widetilde{h}(t)}{t}=+\infty.\label{e2.6} \end{equation} Inequalities \eqref{e2.5}and \eqref{e2.6} imply that there exists $A_\lambda\in(0,\varepsilon)$ such that \begin{equation} \frac{\widetilde{h}(A_\lambda)}{A_\lambda} =\frac{1}{2\lambda\|p\|_\infty}.\label{e2.7} \end{equation} Now, let $(u,v)\in C(\overline{\Omega})\times C(\overline{\Omega})$ and $\theta\in(0,1)$ be such that $(u,v)=\theta A(u,v)$. Then we have \begin{equation} \begin{aligned} \|(u,v)\|&=\max\{\|u\|_\infty,\|v\|_\infty\}\\ &\leq \max\big\{\lambda \|p_1\|_\infty\widetilde{f}(\|v\|_\infty), \lambda \|p_2\|_\infty\widetilde{g}(\|u\|_\infty)\big\}\\ &\leq \max\big\{\lambda \|p_1\|_\infty\widetilde{f}(\|(u,v)\|), \lambda \|p_2\|_\infty\widetilde{g}(\|(u,v)\|)\big\}\\ &\leq \max\big\{\lambda \|p\|_\infty\widetilde{f}(\|(u,v)\|), \lambda \|p\|_\infty\widetilde{g}(\|(u,v)\|)\big\}\\ &\leq \lambda \|p\|_\infty\widetilde{h}(\|(u,v)\|), \end{aligned} \label{e2.8} \end{equation} which implies that $\|(u,v)\|\neq A_\lambda$. Note that $A_\lambda\to0$ as $\lambda\to0$. By Lemma \ref{lem1.1}, $A$ has a fixed point $(\tilde{u}_\lambda,\tilde{v}_\lambda)$ with $\|(\tilde{u}_\lambda,\tilde{v}_\lambda)\|\leq A_\lambda<\varepsilon$. Consequently, from \eqref{e2.2} it follows that \begin{equation} \tilde{u}_\lambda(x)\geq\lambda\delta f(0)p_1(x),\quad x\in\Omega;\quad \tilde{v}_\lambda(x)\geq\lambda\delta g(0)p_2(x),\quad x\in\Omega.\label{e2.9} \end{equation} The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $$ q_1(x)=\int_\Omega G(x,y)a^-(y)dy,\quad q_2(x)=\int_\Omega G(x,y)b^-(y)dy. $$ It follows from (H2), (H3) and Lemma \ref{lem2.1} that there exist four positive constants $\alpha_1, \alpha_2, \gamma_1, \gamma_2\in(0,1)$ such that \begin{gather*} q_1(x)|f(s)|\leq\gamma_1p_1(x)f(0),\quad \text{for } s\in[0,\alpha_1],\; x\in\Omega;\\ q_2(x)|g(s)|\leq\gamma_2p_2(x)g(0),\quad \text{for } s\in[0,\alpha_2],\ x\in\Omega. \end{gather*} Let $\alpha=\min\{\alpha_1, \alpha_2\}$. Then \begin{gather} q_1(x)|f(s)|\leq\gamma_1p_1(x)f(0),\quad \text{for } s\in[0,\alpha],\; x\in\Omega;\label{e2.10}\\ q_2(x)|g(s)|\leq\gamma_2p_2(x)g(0),\quad \text{for } s\in[0,\alpha],\; x\in\Omega.\label{e2.11} \end{gather} Fix $\delta\in(\gamma,1)$, where $\gamma=\max\{\gamma_1,\gamma_2\}$. Let $h(0)=\max\{f(0),g(0)\}$ and let $\lambda_1^\ast, \lambda_2^\ast$ be so small such that \begin{gather*} \|\tilde{u}_\lambda\|_\infty+\lambda\delta h(0)\|p\|_\infty \leq\alpha,\quad \text{for } \lambda\in(0,\lambda_1^\ast),\\ \|\tilde{v}_\lambda\|_\infty+\lambda\delta h(0)\|p\|_\infty\leq\alpha, \quad \text{for } \lambda\in(0,\lambda_2^\ast), \end{gather*} where $\tilde{u}_\lambda$ and $\tilde{v}_\lambda$ are given by Lemma \ref{lem2.1}, and \begin{gather*} |f(t)-f(s)|\leq f(0)\frac{\delta-\gamma_1}{2},\quad \text{for } t, s\in[-\alpha,\alpha],\; |t-s|\leq\lambda_1^\ast \delta h(0)\|p\|_\infty,\\ |g(t)-g(s)|\leq g(0)\frac{\delta-\gamma_2}{2},\quad \text{for } t, s\in[-\alpha,\alpha],\; |t-s|\leq\lambda_2^\ast \delta h(0)\|p\|_\infty. \end{gather*} Let $\lambda^\ast=\min\{\lambda_1^\ast,\lambda_2^\ast\}$. Then for $\lambda\in(0,\lambda^\ast)$, we have \begin{equation} \|\tilde{u}_\lambda\|_\infty+\lambda\delta h(0)\|p\|_\infty\leq\alpha, \quad \|\tilde{v}_\lambda\|_\infty+\lambda\delta h(0)\|p\|_\infty\leq\alpha,\label{e2.12} \end{equation} and for $t, s\in[-\alpha,\alpha]$, $|t-s|\leq\lambda^\ast \delta h(0)\|p\|_\infty$, we have \begin{equation} |f(t)-f(s)|\leq f(0)\frac{\delta-\gamma_1}{2},\quad |g(t)-g(s)|\leq g(0)\frac{\delta-\gamma_2}{2}.\label{e2.13} \end{equation} Now, let $\lambda<\lambda^\ast$. We look for a solution $(u_\lambda,v_\lambda)$ of \eqref{e1.1} of the form $(\tilde{u}_\lambda+m_\lambda,\tilde{v}_\lambda+w_\lambda)$. Thus $(m_\lambda,w_\lambda)$ solves the system \begin{gather*} \Delta m_\lambda=-\lambda a^+(x)(f(\tilde{v}_\lambda+w_\lambda)-f(\tilde{v}_\lambda))+\lambda a^-(x)f(\tilde{v}_\lambda+w_\lambda),\quad \text{in } \Omega,\\ \Delta w_\lambda=-\lambda b^+(x)(g(\tilde{u}_\lambda+m_\lambda)-g(\tilde{u}_\lambda))+\lambda b^-(x)g(\tilde{u}_\lambda+m_\lambda),\quad \text{in } \Omega,\\ m_\lambda=0=w_\lambda.\quad \text{on }\partial\Omega. \end{gather*} For each $(\psi,\varphi)\in C(\overline{\Omega})\times C(\overline{\Omega})$, let $(m,w)=A(\psi,\varphi)$ be the solution of the system \begin{gather*} \Delta m=-\lambda a^+(x)(f(\tilde{v}_\lambda+\varphi)-f(\tilde{v}_\lambda))+\lambda a^-(x)f(\tilde{v}_\lambda+\varphi),\quad \text{in } \Omega,\\ \Delta w=-\lambda b^+(x)(g(\tilde{u}_\lambda+\psi)-g(\tilde{u}_\lambda))+\lambda b^-(x)g(\tilde{u}_\lambda+\psi),\quad\ \text{in } \Omega,\\ m=0=w,\quad \text{on }\partial\Omega. \end{gather*} Then $A:C(\overline{\Omega})\times C(\overline{\Omega})\to C(\overline{\Omega})\times C(\overline{\Omega})$ is completely continuous. Let $(m,w)\in C(\overline{\Omega})\times C(\overline{\Omega})$ and $\theta\in(0,1)$ be such that $(m,w)=\theta A(m,w)$. Then \begin{gather*} \Delta m=-\lambda\theta a^+(x)(f(\tilde{v}_\lambda+w)-f(\tilde{v}_\lambda))+\lambda\theta a^-(x)f(\tilde{v}_\lambda+w),\quad \text{in }\Omega,\\ \Delta w=-\lambda\theta b^+(x)(g(\tilde{u}_\lambda+m)-g(\tilde{u}_\lambda))+\lambda\theta b^-(x)g(\tilde{u}_\lambda+m),\quad \text{in }\Omega,\\ m=0=w,\quad \text{on }\partial\Omega. \end{gather*} Now, we claim that $\|(m,w)\|\neq\lambda\delta h(0)\|p\|_\infty$. Suppose to the contrary that $\|(m,w)\|=\lambda\delta h(0)\|p\|_\infty$, then there are three possible cases. \textbf{Case 1.} $\|m\|_\infty=\|w\|_\infty=\lambda\delta h(0)\|p\|_\infty$. Then we have from \eqref{e2.12} that $\|\tilde{v}_\lambda+w\|_\infty\leq\|\tilde{v}_\lambda\|_\infty +\lambda\delta h(0)\|p\|_\infty\leq\alpha$, and so $\|\tilde{v}_\lambda\|_\infty\leq\alpha$. Thus by \eqref{e2.13} we obtain \begin{equation} |f(\tilde{v}_\lambda+w)-f(\tilde{v}_\lambda)| \leq f(0)\frac{\delta-\gamma_1}{2}.\label{e2.14} \end{equation} On the other hand, \eqref{e2.14} implies \begin{align*} |m(x)| &\leq\lambda p_1(x)f(0)\frac{\delta-\gamma_1}{2} +\lambda\gamma_1p_1(x)f(0)\\ &=\lambda p_1(x)f(0)\frac{\delta+\gamma_1}{2}\\ &<\lambda p_1(x)f(0)\delta\\ &\leq \lambda\delta h(0)\|p\|_\infty,\quad \text{for }x\in\Omega, \end{align*} which implies that $\|m\|_\infty<\lambda\delta h(0)\|p\|_\infty$, a contradiction. \textbf{Case 2.} $\|w\|_\infty<\|m\|_\infty=\lambda\delta h(0)\|p\|_\infty$. Then $\|\tilde{v}_\lambda+w\|_\infty<\|\tilde{v}_\lambda\|_\infty +\lambda\delta h(0)\|p\|_\infty\leq\alpha$, and so $\|\tilde{v}_\lambda\|_\infty\leq\alpha$. Thus $$ |f(\tilde{v}_\lambda+w)-f(\tilde{v}_\lambda)| \leq f(0)\frac{\delta-\gamma_1}{2}. $$ By the same method used to prove Case 1, we can show that $\|m\|_\infty<\lambda\delta h(0)\|p\|_\infty$, which is a desired contradiction. \textbf{Case 3.} $\|m\|_\infty<\|w\|_\infty=\lambda\delta h(0)\|p\|_\infty$. As in Case 2, we obtain $\|w\|_\infty<\lambda\delta h(0)\|p\|_\infty$, a contradiction. Then the claim is proved. By Lemma \ref{lem1.1}, $A$ has a fixed point $(m_\lambda,w_\lambda)$ with $\|(m_\lambda,w_\lambda)\|\leq\lambda\delta h(0)\|p\|_\infty$. Using Lemma \ref{lem2.1}, we obtain \begin{align*} u_\lambda(x) &\geq\tilde{u}_\lambda(x)-|m_\lambda(x)|\\ &\geq \lambda\delta p_1(x)f(0) -\lambda\frac{\delta+\gamma_1}{2}f(0)p_1(x)\\ &=\lambda\frac{\delta-\gamma_1}{2}f(0)p_1(x)\\ &>0,\quad x\in\Omega. \end{align*} Similarly, we can prove that $v_\lambda(x)>0, x\in\Omega$. The proof is complete. \end{proof} \begin{proof}[Proof of Corollary \ref{coro1.1}] Multiplying the both sides of the equation \begin{equation} u''+\frac{N-1}{t}u'=-a^\pm(t),\quad u'(0)=u(1)=0\label{e2.15} \end{equation} by $t^{N-1}$, we obtain \begin{equation} (t^{N-1}u')'=-a^\pm(t)t^{N-1}.\label{e2.16} \end{equation} Integrating the both sides of \eqref{e2.16} from $0$ to $t$, we have $$ t^{N-1}u'(t)=-\int_0^ta^\pm(s)s^{N-1}ds. $$ Integrating the both sides of above equation from $t$ to $1$, we have \begin{equation} u^\pm(t)=\int_t^1\frac{1}{s^{N-1}} \Big(\int_0^sa^\pm(\tau)\tau^{N-1}d\tau\Big)ds.\label{e2.17} \end{equation} Therefore the solution of problem \eqref{e2.15} is given by \eqref{e2.17}. This implies that $u^+\geq k_1u^-$. By the same method, we can show that $v^+\geq k_2v^-$, and the result follows from Theorem \ref{thm1.1}. \end{proof} \section{$n\times n$ systems} In this section, we consider the existence of positive solutions of the $n\times n$ system \begin{equation} \begin{gathered} -\Delta u_1=\lambda a_1(x)f_1(u_2),\quad \text{in }\Omega,\\ -\Delta u_2=\lambda a_2(x)f_2(u_3),\quad \text{in }\Omega,\\ \cdots\\ -\Delta u_{n-1}=\lambda a_{n-1}(x)f_{n-1}(u_n),\quad \text{in }\Omega,\\ -\Delta u_n=\lambda a_n(x)f_n(u_1),\quad \text{in }\Omega,\\ u_1=u_2=\cdots=u_n=0,\quad \text{on }\partial\Omega, \end{gathered}\label{e3.1} \end{equation} where $a_i\in L^\infty(\Omega)\ (i=1,2,\dots,n)$ may be sign-changing in $\Omega$ and $\lambda>0$ is a parameter. We assume the following conditions: \begin{itemize} \item[(H4)] $f_i: [0,\infty)\to\mathbb{R}$ which is continuous and $f_i(0)>0\ (i=1,2,\dots,n)$; \item[(H5)] $a_i\ (i=1,2,\dots,n)$ is continuous on $\overline{\Omega}$ and there exists $k_i>1\ (i=1,2,\dots,n)$ such that $$ \int_{\Omega}G(x,y)a_i^+(y)dy\geq k_i\int_{\Omega}G(x,y)a_i^-(y)dy, \quad \forall x\in\Omega, $$ where $G(x,y)$ is defined as in Section 2. \end{itemize} Define the integral equation $$ (u_1,u_2,\dots,u_n)=A(u_1,u_2,\dots,u_n), $$ where $A:(C(\overline{\Omega}))^n\to (C(\overline{\Omega}))^n$ is defined by \begin{align*} &A(u_1,u_2,\dots,u_n)(x)\\ &=\Big(\lambda\int_\Omega G(x,y)a_1(y)f_1(u_2)dy,\dots,\lambda\int_\Omega G(x,y)a_n(y)f_n(u_1)dy \Big). \end{align*} \begin{theorem} \label{thm3.1} Let {\rm (H4), (H5)} hold. Then there exists a positive number $\lambda^\ast$ such that \eqref{e3.1} has a positive solution for $0<\lambda<\lambda^\ast$. \end{theorem} As a consequence of the above theorem we have the following corollary. \begin{corollary} \label{coro3.1} Let $f_i$ $(i=1,2,\dots,n)$ satisfy {\rm (H4)}. Let $a_i$ $(i=1,2,\dots,n)$ be nonzero integrable functions on $[0,1]$. Suppose that there exist positive constants $k_i>1$ such that $$ \int_0^ts^{N-1}a_i^+(s)ds\geq k_i\int_0^ts^{N-1}a_i^-(s)ds,\quad \text{for } t\in[0,1],\; (i=1,2,\dots,n). $$ Then there exists a positive number $\lambda^\ast$ such that the system \begin{gather*} u_1''+\frac{N-1}{t}u_1'+\lambda a_1(t)f_1(u_2)=0,\quad 0