\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). {\em Electronic Journal of Differential Equations}, Conference 21 (2014), pp. 23--30. ISSN: 1072-6691. http://ejde.math.txstate.edu, http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{23} \title[\hfilneg EJDE-2014/Conf/21 \hfil Existence of positive solutions] {Existence of positive solutions for a superlinear elliptic system with Neumann boundary condition} \author[J. C. Carde\~no, A. Castro \hfil EJDE-2014/Conf/21\hfilneg] {Juan C. Carde\~no, Alfonso Castro} % in alphabetical order \address{Juan C. Carde\~no \newline Facultad de Ciencias Naturales y Matem\'aticas, Universidad de Ibagu\'e, Ibagu\'e, Tolima, Colombia} \email{juan.cardeno@unibague.edu.co} \address{Alfonso Castro \newline Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA} \email{castro@math.hmc.edu} \thanks{Published February 10, 2014.} \subjclass[2000]{35J20, 35J60, 35B38} \keywords{Semilinear elliptic system; a priori estimates; ordered Banach space; \hfill\break\indent contraction-expansion operator; Neumann boundary condition} \begin{abstract} We prove the existence of a positive solution for a class of nonlinear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} The purpose of this paper is to prove that the system \begin{equation}\label{sistema} \begin{gathered} -\Delta u+ \alpha u =\beta v+f_{1}(x,u,v)\quad \text{in }\Omega \\ -\Delta v+\delta v = \gamma u+f_2(x,u,v)\quad \text{in }\Omega \\ \frac{\partial u }{ \partial n} = \frac{\partial v }{ \partial n} =0 \quad \text{in } \partial \Omega, \end{gathered} \end{equation} has a nontrivial positive solution. In \eqref{sistema} $\Delta$ denotes the Laplacian operator, $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, and $\alpha >0 $, $\beta >0 $, $\gamma >0 $, $\delta >0$ are real parameters. We also assume that $f_1(x,u,v), f_2(x,u,v)$ are measurable in $x$, differentiable in $(u,v)$, and bounded on bounded sets. Our main result reads as follows. \begin{theorem}\label{main} If there exist $b \in (1, \min\{2, (N+1)/(N-1)\}) $, $m>0$, and $M>0$ such that \begin{equation}\label{b} m(u+v)^b \leq f_{i}(x,u,v) \leq M (u+v)^b \quad \text{for } i= 1,2, u,v \geq 0, \end{equation} and $\beta \gamma < \alpha \delta$, then the problem \eqref{sistema} has a positive solution. \end{theorem} The main tool in our proofs is Krasnoselskii's compression-expansion theorem (see Theorem \ref{Krasno} below) which we state for sake of completeness. For a proof of this theorem the reader is referred to \cite[Theorem 13.D]{zeidler}. To apply Theorem \ref{Krasno} to Theorem \ref{main}, in Section 3 we use of a priori estimates for elliptic equation with Neumann boundary conditions, see \cite{zanger}. \begin{theorem}\label{Krasno} Let $X$ be a real ordered Banach space with positive cone $K$. If $\Upsilon: K \to K$ is a compact operator and there exist real numbers $0 0$, $\beta > 0$, $\gamma > 0$, and $\delta > 0$. \begin{lemma}\label{lemau(v)} For each $P_{1}, v\in L^{2}(\Omega)$, then the equation \begin{equation}\label{u(v)} \begin{gathered} -\Delta u+\alpha u = P_{1}(x)+ \beta v \quad \text{in }\Omega \\ \frac{\partial u }{\partial n}=0\quad \text{in } \partial \Omega, \end{gathered} \end{equation} has a unique solution. Moreover, there exists $c>0$, independent of $(P_1, v)$, such that \begin{equation} \|u\|_{1,2} \leq c\|P_1 + \beta v\|_2 , \end{equation} \end{lemma} \begin{proof} Let $H$ be the Sobolev space $H^{1}(\Omega)$, and $B:H \times H \to \mathbb{R}$ defined by $B[u,v]=\int_{\Omega}{\nabla u \nabla v+ \alpha uv}$. Since $\alpha > 0$, $B[u,u]\geq \min \{1,\alpha\}\|u\|^{2}$. By the Lax-Milgram theorem (see \cite{evans}) there exists $u\in H $ such that \begin{equation} B[u,z]=\int_{\Omega}{\nabla u \nabla z}+ \alpha \int_{\Omega}{uz} = \int_{\Omega}z(x)(P_{1}(x)+ \beta v(x) )dx. \end{equation} Hence $u$ is a weak solution to \eqref{u(v)}. Taking $z=u$ and $c^{-1} = \min\{1, \alpha\}$ the lemma is proved. \end{proof} \begin{lemma}\label{upositiva} Let $P_{1}$, $v$, and $u $ be as in Lemma \ref{lemau(v)}. If $v\geq 0$ then $u\geq 0$. \end{lemma} \begin{proof} Suppose $u$ is not positive. Let $A=\{x\in\Omega, u(x)< 0 \}$, and $z= u \chi_{A}$. By the definition of weak solution \begin{equation} \int_{\Omega}{z(P_{1}+ \beta v)} = \int_{\Omega}{\nabla u \nabla z}+ \alpha \int_{\Omega}{uz} = \int_{A}{\nabla u \nabla u}+ \alpha (\int_{A}{u^{2}}). \end{equation} This is a contradiction since $\int_{A}{\nabla u \nabla u} + \alpha (\int_{A}{u^{2}}) >0$, while $\int_{A}{z(P_{1}+ \beta v)} <0$. This proves the lemma. \end{proof} \begin{lemma}\label{lemaw(u(v))} For each $v\in L^{2}$, let $u(v) \equiv u \in H^{1}(\Omega)$ be the solution to \eqref{u(v)} given by Lemma \ref{lemau(v)}. If $w\in H^{1}(\Omega)$ is the weak solution to \begin{equation}\label{w(u(v))} \begin{gathered} -\Delta w +\delta w = P_2(x)+ \gamma u(v) \quad \text{in }\Omega \\ \frac{\partial w }{\partial n}=0 \quad \text{in } \partial \Omega, \end{gathered} \end{equation} then \begin{equation} \|w\|_2\leq\frac{1}{\alpha} \|P_2\|_2 + \frac{\delta}{\alpha\gamma}\|P_{1}\|_2 +\frac{\beta\gamma}{\delta\alpha}\|v\|_2. \end{equation} \end{lemma} \begin{proof} Multiplying \eqref{w(u(v))} by $w$ and using the Cauchy-Schwartz inequality we have \begin{equation}\label{intw} \begin{aligned} \int_{\Omega}{\nabla w \nabla w}+ \delta \int_{\Omega}{w^{2}} &= \int_{\Omega}{P_2(x)\cdot w + \gamma u(v)}\cdot w\\ &\leq \|P_2\|_2 \cdot \|w\|_2 + \gamma\|u(v)\|_2 \cdot \|w\|_2 \\ &\leq (\|P_2\|_2 + \delta\|u(v)\|_2)\cdot \|w\|_2. \end{aligned} \end{equation} Hence \begin{equation}\label{w<} \|w\|_2 \leq \frac{1}{\delta} \|P_2\|_2 + \frac{\gamma}{\delta} \|u(v)\|_2. \end{equation} Similarly, \begin{equation}\label{u<} \|u\|_2 \leq \frac{1}{\alpha} \|P_{1}\|_2 + \frac{\beta}{\alpha} \|v\|_2. \end{equation} Replacing \eqref{w<} in \eqref{u<}, \begin{equation} \begin{aligned} \|w\|_2 & \leq \frac{1}{\gamma} \|P_2\|_2 + \frac{\gamma}{\delta} \|u(v)\|_2 \\ & \leq \frac{1}{\gamma} \|P_2\|_2 + \frac{\delta}{\gamma}(\frac{1}{\alpha} \|P_{1}\|_2 +\frac{\beta}{\alpha}\|v\|_2) \\ & \leq \frac{1}{\alpha} \|P_2\|_2 + \frac{\delta}{\alpha\gamma}\|P_{1}\|_2 +\frac{\beta\gamma}{\delta\alpha}\|v\|_2, \end{aligned} \end{equation} which proves the lemma. \end{proof} \begin{theorem}\label{uvexiste} Given $(P_{1}, P_2) \in L^{2}(\Omega)\times L^{2}(\Omega)$, there exists a unique pair $(u,v)\in H \times H $ satisfying \eqref{sistemalineal}. In addition, $(u,v)$ depends continuously on $(P_{1},P_2)$. \end{theorem} \begin{proof} Let $v_{1},v_2\in L^{2}(\Omega).$ Let $u(v_{1})$ and $u(v_2)$ be given by Lemma \ref{lemau(v)} and $w_{1}, w_2$ as given by Lemma \ref{lemaw(u(v))}. Hence \begin{equation} \begin{aligned} &\int_{\Omega}|{\nabla (w_{1}-w_2)|^{2}} + \delta \int_{\Omega}{|(w_{1}-w_2)|^{2}}\\ &= \gamma \int_{\Omega}{u(v_{1})-u(v_2))(w_{1}-w_2)} \\ & \leq \gamma(\|u(v_{1})-u(v_2))\|_{L_2})\|w_{1}-w_2)\|_2. \end{aligned} \end{equation} Therefore, \begin{equation}\label{w1-w2} \|w_{1}-w_2\|\leq \frac{\gamma}{\delta}(\|u(v_{1})-u(v_2))\|_{L_2}. \end{equation} Multiplying \eqref{u(v)} by $u(v_{1})-u(v_2)$ and subtracting we have \begin{equation} \begin{aligned} &\int_{\Omega}|\nabla(u_{1}-u_2)|^{2} + \alpha \int_{\Omega}(u(v_{1})-u(v_2))^{2} \\ &= \beta\int_{\Omega}((v_{1}-v_2)(u(v_{1})-u(v_2)) \\ & \leq {\beta}\|v_{1}-v_2\|_2\|u(v_{1})-u(v_2)\|_2. \end{aligned} \end{equation} Thus \begin{equation} \|(u(v_{1})-u(v_2)\|_2\leq \frac{\beta}{\alpha}\|(v_{1}-v_2)\|_2. \end{equation} Replacing this in \eqref{w1-w2} yields $\|w_{1}-w_2\|_2\leq \frac{\gamma\beta}{\alpha\delta}\|(v_{1}-v_2)\|_2$. Hence by the contraction mapping principle there exists a unique $w$ such that $w=v$. That is $(u,v)$ satisfies \begin{equation} \begin{gathered} -\Delta u +\alpha u = \beta v + P_{1}(x) \quad\text{in }\Omega \\ -\Delta v +\delta v = \gamma u + P_2(x) \quad\text{in }\Omega \\ \frac{\partial u }{\partial n}= 0 =\frac{\partial v }{\partial n} \quad \text{on }\ \partial \Omega, \end{gathered} \end{equation} By Lemma \ref{lemau(v)}, $u$ depends continuously on $(P_1, v)$. Also, by Lemma \ref{lemaw(u(v))}, $v$ depends continuously on $(P_1, P_2)$. Hence $(u,v)$ depends continuously on $(P_1, P_2)$, which proves the theorem. \end{proof} \begin{lemma}\label{lemC2} Let $h_1, h_2 \in L^{\infty}(\Omega)$. For each $p > 1 $ there exist $C_2(p)\equiv C_2 > 0$ such that if $(y,z)$ satisfies \begin{equation}\label{agdoni} \begin{gathered} -\Delta y +\alpha y = \beta z + h_1, \\ -\Delta z +\delta z = \gamma y + h_2, \quad\text{in }\Omega \\ \frac{\partial y }{\partial n} = \frac{\partial z }{\partial n} = 0 \quad \text{in } \partial \Omega, \end{gathered} \end{equation} then \begin{equation}\label{C2} \|y\|_{2,p} + \|z\|_{2,p} \leq C_2 (\|h_1\|_{\infty} + \|h_2\|_{\infty}) \end{equation} (see \cite{evans}). In particular, by the Sobolev imbedding theorem, taking $p >N/2$ we may assume that \begin{equation} \|y\|_{\infty} + \sup\frac{|y(\zeta) - y(\eta)|}{\|\zeta - \eta\|} + \|z\|_{\infty} + \sup\frac{|z(\zeta) - z(\eta)|}{\|\zeta - \eta\|} \leq C_2 (\|h_1\|_{p} + \|h_2\|_{p}). \end{equation} \end{lemma} \begin{proof} Multiplying the first equation in \eqref{agdoni} by $y$ we have \begin{equation}\label{pory} \begin{aligned} \int_{\Omega}|\nabla y|^{2}+ \alpha \int_{\Omega}y^{2} &=\beta \int_{\Omega}( yz)+ \int_{\omega}{h_{1}y} \\ & \leq \beta\int_{\Omega}(yz)+ \|h_1\|_{\infty}|\Omega|^{1/2}\|y\|_2. \end{aligned} \end{equation} Similarly, \begin{equation}\label{porz} \begin{aligned} \int_{\Omega}|\nabla z|^{2}+ \delta \int_{\Omega}z^{2} &=\gamma \int_{\Omega}( yz)+ \int_{\omega}{h_2z} \\ & \leq \gamma \int_{\Omega}(yz)+ \|h_2\|_{\infty}|\Omega|^{1/2}\|z\|_2. \end{aligned} \end{equation} Since $\alpha > 0 $ and $\alpha \delta - \beta \gamma >0$, the quadratic form $G(s,t)= \alpha s^2 -(\beta +\gamma)st +\delta t^2$ positive definite. That is, there exists $C>0$ such that $G(s,t) \geq C(s^2 + t^2)$ for all $s,t \in \mathbb{R}$. This, \eqref{pory}, and \eqref{porz} imply \begin{equation}\label{defC} C(\|y\|_2 + \|z\|_2) \leq 2|\Omega|^{1/2}(\|h_1\|_{\infty} + \|h_2\|_{\infty}). \end{equation} By \eqref{pory} and \eqref{defC}, \begin{equation} \begin{aligned} \bar \alpha \|y\|_{1,2}^2 & \leq \|y\|_2 (\beta \|z\|_2 + |\Omega|^{1/2}( \|h_1\|_{\infty}+ \|h_2\|_{\infty})) \\ & \leq (\frac{2\beta }{C} + 1)|\Omega|^{1/2} \|y\|_2 ( \|h_1\|_{\infty}+ \|h_2\|_{\infty}) \\ & \equiv C_3 \|y\|_2 ( \|h_1\|_{\infty}+ \|h_2\|_{\infty}) \\ & \leq C_3 \|y\|_{{1,2} }( \|h_1\|_{\infty}+ \|h_2\|_{\infty}). \end{aligned} \end{equation} Hence \begin{equation}\label{yenH1} \|y\|_{1,2} \leq \frac{C_3}{\bar \alpha }(\|h_1\|_{\infty}+ \|h_2\|_{\infty}). \end{equation} Similarly, \begin{equation}\label{zenH1} \|z\|_{1,2} \leq \frac{C_3}{\bar \delta }(\|h_1\|_{\infty}+ \|h_2\|_{\infty}). \end{equation} From \eqref{yenH1}, \eqref{zenH1} and the Sobolev imbedding theorem (see \cite[Theorem ??]{evans}) we see that \begin{equation}\label{esty+z} \begin{aligned} \|y\|_{2N/(N-2)} + \|z\|_{2N/(N-2)} & \leq S(1,2)(\|y\|_{1,2} + \|z\|_{1,2}) \\ & \leq S(1,2)\Big(\frac{C_3}{\bar \alpha } + \frac{C_3}{\bar \delta }\Big)(\|h_1\|_{\infty}+ \|h_2\|_{\infty}) \\ & \equiv C_4(\|h_1\|_{\infty}+ \|h_2\|_{\infty}). \end{aligned} \end{equation} By regularity properties for elliptic boundary value problems there exists a positive real number $C_2$ such that if $-\Delta u + \tau u = f$ en $\Omega$ and $(\partial u)/(\partial \eta) = 0$ in $\partial \Omega$ $\|u\|_{2,P}$ when $p \in (1, (N/2) + 1)$. This and \eqref{esty+z} imply \begin{equation}\label{esty+zW2p} \|y\|_{2,\frac{2N}{N-2}}+ \|z\|_{2,\frac{2N}{N-2}} \leq C_2(C_4 + |\Omega|^{\frac{N-2}{2N}})(\|h_1\|_{\infty}+ \|h_2\|_{\infty})). \end{equation} Iterating this argument finitely many times we see that there exist $p>N/2$ and $C_3>0$ such that \begin{equation}\label{esty+zW2pA} \|y\|_{2,p}+ \|z\|_{2,p} \leq C_3(\|h_1\|_{\infty}+ \|h_2\|_{\infty})), \end{equation} which proves the lemma. \end{proof} \section{Proof of Theorem \ref{main}} Let $\rho = \max\{\alpha/m, \delta/m\}$ and $\bar R = 2(2M\rho |\Omega|)^{1/(2-b)}$ (see \eqref{b}). For $i = 1,2$, let \[ g_i(x,u,v) =\begin{cases} f_i(x,u, v) &\text{for }0 \leq u+v \leq \bar R, \\ f_i(x,\bar R u/(u+v),\bar R v/(u+v)) &\text{for }u+v \geq \bar R. \end{cases} \] Let $X$ be the ordered Banach space $C(\bar \Omega) \times C(\bar \Omega)$ with positive cone \begin{equation} \begin{aligned} K = \Big\{&(u,v) \in X: u \geq 0, v \geq 0, \|u - \frac{1}{|\Omega|} \int_{\Omega} u \|_{\infty} \leq b M\bar R^{b-1} \int_{\Omega} u, \\ & \|v - \frac{1}{|\Omega|} \int_{\Omega} v \|_{\infty} \leq b M\bar R^{b-1} \int_{\Omega} v \Big\}. \end{aligned} \end{equation} Let (see \eqref{b} and Lema \ref{lemC2}) \begin{equation}\label{R} R \in \big(0, \min\{\bar R, (2C_2M)^{1-b}\}\big). \end{equation} For $(u,v) \in K$, $\|(u,v)\|_X \geq R$, we define $\Upsilon(u,v) = (U,V)$ as the only solution to \begin{equation}\label{UV} \begin{gathered} -\Delta U +\alpha U = \beta V + g_1(x,u,v) \quad\text{in }\Omega \\ -\Delta V +\delta V = \gamma U+ g_2(x,u,v) \quad\text{in }\Omega \\ \frac{\partial u }{\partial n}= 0 =\frac{\partial v }{\partial n} \quad \text{in } \partial \Omega. \end{gathered} \end{equation} If $(u,v) \in K$ and $\|(u,v)\|_X \leq R$ we define \begin{equation} \Upsilon(u,v) = \|(u,v)\|_X \Upsilon((R/ \|(u,v)\|_X )(u,v)), \quad \Upsilon(0,0) = (0.0). \end{equation} Since $g_1$, $g_2$ are nonnegative continuous functions, $\Upsilon(u,v) = (U,V)$ satisfies $U \geq 0$ y $V\geq 0$ for $(u,v) \in K$ (see Lemma \ref{upositiva}). Suppose that for some $(U, V) = \Upsilon(u,v)$ we have \begin{equation}\label{contradU} \|U - \frac{1}{|\Omega|} \int_{\Omega} U \|_{\infty} > b M\bar R^{b-1} \int_{\Omega} U, \end{equation} with $\|(u,v)\|_X \geq R$. Hence $\|U\|_{\infty} \geq bMR^{b-1}\int_{\Omega} U$, which implies that if $\|U\|_{\infty} = U(x)$, $x \in \bar \Omega$, then there exists $y \in \bar \Omega$ such that $\|y -x\| \leq m_1 \bar R^{(1-b)/n}$ and $U(y) \leq U(x)/2$, with $m_1$ a constant depending only on $\Omega$. Hence \begin{equation} \frac{U(x) - U(y)}{\|x - y\|} \geq \frac{\|U\|_{\infty}}{2m_1\bar R^{(b-1)/N}}. \end{equation} Let now $p>N$ be such that \begin{equation} \frac{N+p - b(p-1)}{(p-1)N} + \frac{b}{p} > 0. \end{equation} This and Lemma \ref{lemC2} imply \begin{equation}\label{Uinf1} \begin{aligned} \|U\|_{\infty} \bar R^{(b-1)/n} & \leq C_2 \|g_1(\cdot,u,v)\|_p \\ & \leq C_2M\Big(\int_{\Omega} (u+v)^{bp}\Big)^{1/p} \\ & \leq C_2M\Big(\int_{\Omega} (u+v)^b(u+v)^{b(p-1)}\Big)^{1/p} \\ & \leq C_2M\|u+v\|_{\infty}^{b(p-1)/p} \Big(\int_{\Omega} (u+v)^b\Big)^{1/p}. \end{aligned} \end{equation} Integrating the first equation in \eqref{UV} on $\Omega$, \begin{equation}\label{UL1>} \alpha \int_{\Omega} U \geq m\int_{\Omega} (u+v)^b, \end{equation} (see \eqref{b}). From \eqref{Uinf1} and \eqref{UL1>}, \begin{equation} \begin{aligned} \|U\|_{\infty} \bar R^{\frac{b-1}{n}} &\leq C_2M\|u+v\|_{\infty}^{b(p-1)/p} \Big(\frac{\alpha}{m}\int_{\Omega} U \Big)^{1/p} \\ & \leq C_2M\|u+v\|_{\infty}^{b(p-1)/p} \Big(\frac{\alpha}{2mM}\bar R^{1-b} \|U\|_{\infty} \Big)^{1/p} \\ & \leq C_2M\Big(2M \bar R^{b-1} \int_{\Omega} (u+v)\Big)^{\frac{b(p-1)}{p}} \Big(\frac{\alpha}{2mM}\bar R^{1-b}\int_{\Omega} \|U\|_{\infty} \Big)^{1/p}. \end{aligned} \end{equation} Therefore \begin{equation} \begin{aligned} \|U\|_{\infty}^{(p-1)/p} & \leq m_2 \bar R^{(b-1)(\frac{b(p-1)}{n} -\frac1p - \frac1n)}\Big( \int_{\Omega} (u+v)\Big)^{b(p-1)/p}\\ & \leq m_3 \bar R^{(b-1)(\frac{b(p-1)}{n} -\frac1p - \frac1n)} \Big( \int_{\Omega} (u+v)^b\Big)^{(p-1)/p} \\ & \leq m_3 \bar R^{(b-1)(\frac{b(p-1)}{n} -\frac1p - \frac1n)} \Big( \frac{\alpha}{m}\int_{\Omega} U\Big)^{(p-1)/p} \\ & \leq m_4 \bar R^{(b-1)(\frac{b(p-1)}{n} -\frac1p - \frac1n)} \Big( \bar R^{1-b} \| U\|_{\infty}\Big)^{(p-1)/p}. \end{aligned} \end{equation} Since $m_2$, $m_3$, $m_4$ are independent of $U$, \begin{equation} 1 \leq m_4 \bar R^{(b-1)(\frac{b(p-1)}{n} -\frac1p - \frac1n -\frac{p-1}{p})}. \end{equation} By \eqref{b}, there exists $p>N$ such that \begin{equation}\label{exp<0} (b-1)\Big(\frac{b(p-1)}{n} -\frac1p - \frac1n -\frac{p-1}{p}\Big) < 0. \end{equation} Taking $\bar R$ sufficiently large we have a contradiction to \eqref{contradU}. Thus $\Upsilon(u,v) \in K$. For $\|(u,v)\|_X < R$ the proof follows from the definition of $\Upsilon$. Thus $\widehat{}\Upsilon(K) \subset K$. Let $C_2$ be as in \ref{lemC2} and $x \in \bar \Omega$ be such that $U(x) = \max\{U(y); y \in \bar \Omega\}$. From the definition of $C_2$ we conclude that if $y \in \bar \Omega$ and $\|y-x\|\leq C_2M (\|u\|_{\infty}^b + \|v\|_{\infty}^b)$ then by the definition of $g_1, g_2$, if $\{u_j,v_j\}_j$ is a bounded sequence in $X$ so are $\{g_1(x,u_j,v_j)\}_j$ and $\{g_2(x,u_j,v_j)\}_j$ in $C(\bar \Omega)$. Since $g_1, g_2$ are bounded functions, due to Lemmas \ref{lemC2}, $\{U_j,V_j\}_j$ is bounded in $W^{2,p}(\Omega) \times W^{2,p}(\Omega)$. Taking $p > N/2$, by the Sobolev imbedding theorem (see \cite{evans}) we see that $\{U_j,V_j\}_j$ has a converging subsequence in the space $X$, which proves that $\Upsilon$ is a compact operator. Suppose that for some $(u,v)$ such that $\|u\|_{\infty} + \|v\|_{\infty} = R$, $U\geq u$, $V\geq v$. By \eqref{C2}, \begin{equation} \begin{aligned} R & = \|u\|_{\infty} + \|v\|_{\infty} \leq \|U\|_{\infty} + \|V\|_{\infty}\\ & \leq 2C_2M\|u+v\|_{\infty}^b \\ & \leq 2C_2MR^b, \end{aligned} \end{equation} which contradicts the definition of $R$. This proves that $\Upsilon(u,v) \not \geq (u,v)$ for $\|(u,v)\|_X\\ = R$. Suppose that $(U,V) = \Upsilon(u,v) \leq (u,v)$ for some $(u,v)$ with $\|(u,v)\|_X = \bar R$. Without loss of generality we may assume that $\|u\| \geq \bar R/2$. Hence, by the definition of $K$, \begin{equation} \int_{\Omega} u \geq \bar R\frac{1}{2(|\Omega|^{-1} + bM\bar R^{b-1})} \geq C_3 \bar R^{2 -b}. \end{equation} Integrating the first equation in \eqref{UV} we infer that \begin{equation}\label{intU+V<} \begin{aligned} \alpha \int_{\Omega}U & =\beta \int_{\Omega}V + \int_{\Omega}g_1(u,v) \\ & = \beta \int_{\Omega}V + m \int_{\Omega}(u+v)^{b} \\ & \geq \beta \int_{\Omega}V + m \int_{\Omega}(U+V)^{b}. \end{aligned} \end{equation} Similarly, \[ \delta \int_{\Omega}V \geq \gamma \int_{\Omega}U +m \int_{\Omega}(U+V)^{b}. \] By Holder inequality and the definition of $\rho$, \begin{equation}\label{intU+V} \int_{\Omega}(U + V)^b \leq \rho |\Omega|. \end{equation} Since $(U,V) \in K$, \begin{equation} \bar R \leq 2\|U\|_{\infty} \leq 4MR^{b-1}\int_{\Omega} U \leq 2M\bar R^{b-1}\rho |\Omega|, \end{equation} which contradicts the definition of $\bar R$. Thus $\Upsilon$ satisfies the hypotheses of Theorem \ref{Krasno}. Hence $\Upsilon$ has a fixed point $(u,v)$ in $\{(y,z); \|(y,z)\|\in (R\ ,\ \overline{R})$. Therefore $(u,v)$ is a positive solution to \eqref{sistema}, which proves Theorem \ref{main}. \subsection*{Acknowledgments} Alfonso Castro was partially supported by a grant from the Simons Foundations (\# 245966). \begin{thebibliography}{00} \bibitem{adams} Adams, R.; \emph{Sobolev Space, second edition}, Academic Press, (2003) \bibitem{an} An, Y.; Fan, X.; \emph{On the coupled systems of second and fourth order elliptic equation}, Applied Mathematics and Computation, (2003). \bibitem{costa} Costam D. G.; Magalhaes, C. A.; \emph{A Variational appoach to subquadratic perturbations of elliptic systems,} Journal of Differential Equations, vol 111 ( 1994), 103-121. \bibitem{figueiredo} De Figueiredo, D.; \emph{Semilinear elliptic systems,} Segundo Congreso Latinoamericano de Matematicas, Cancun Mexico (2004). \bibitem{evans} Evans, L. C.; \emph{Partial Differential Equations: Second Edition}, University of California, Berkeley-AMS, (2010). \bibitem{kwong} Kwong M.K; \emph{On Krasnoselskii's cone fixed point theorem}, Lucent Technologies, Inc USA. \bibitem{oregan} O'Regan, D.; \emph{A Krasnoselskii cone compression result for multimaps in the S-KKM class}, L'Institut Mathematique Nouvelle Serie, Tome 77, 2005. \bibitem{qouc} Qouc, A. N.; \emph{An application of the Lyapunov-Schmidt method to semilinear elliptic problems}, Electronic Journal of Differential Equations, No. 129, pp. 1-11, (2005). \bibitem{rabinowitz} Rabinowitz, P. H.; \emph{Minimax methods in critical point theory and applications to differential equations}, Regional Conference Series in Mathematics, Number 65, AMS, Providence R.I., (1986). \bibitem{soares} Soarez dos Santos, J.; \emph{Soluci\'on de un sistema de ecuaciones diferenciales elipticas via teoria de puntos fijos en conos,} Universidad Federal de Campina Grande Brasil, (2007). \bibitem{zanger} Zanger, Z. D.; \emph{The inhomogeneous Neumann problem in Lipschitz domains}, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1771-1808. \bibitem{zeidler} Zeidler, E.; \emph{Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems}, Springer- Verlag, (1986). \end{thebibliography} \end{document}