\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 134, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/134\hfil Existence and uniqueness] {Existence and uniqueness of solutions to a semilinear elliptic system} \author[Z. Chen, Y. Cui\hfil EJDE-2009/134\hfilneg] {Zu-Chi Chen, Ying Cui} % in alphabetical order \address{Zu-Chi Chen \newline Department of mathematics\\ University of Science and Technology of China\\ Hefei 230026, China} \email{chenzc@ustc.edu.cn} \address{Ying Cui \newline Department of mathematics\\ University of Science and Technology of China\\ Hefei 230026, China} \email{cuiy@mail.ustc.edu.cn} \thanks{Submitted July 6, 2009. Published October 19, 2009.} \thanks{Supported by grant 10371116 from the NNSF of China} \subjclass[2000]{35J55, 35J60, 35J65} \keywords{Super-sub solutions; compact continuous operator; \hfill\break\indent Leray-Schauder fixed point theorem} \begin{abstract} In this article, we show the existence and uniqueness of smooth solutions for boundary-value problems of semilinear elliptic systems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction and main results} We study the solvability for the semilinear elliptic system with homogeneous Dirichlet boundary value condition \begin{equation} \label{e1.1} \begin{gathered} L_1 u= f(x,u,v,Du,Dv), \quad x\in \Omega \\ L_2 v= g(x,u,v,Du,Dv), \quad x\in \Omega \\ u = v = 0, \quad x\in \partial\Omega \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^N$ ($N\geq 2$) denotes a bounded domain with smooth boundary, and $f,g: \overline{\Omega}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $L_1$ and $L_2$ are the uniformly elliptic operators of second order: $$ L_ku=\sum_{i,j=1}^N \partial_{x_{j}}(a_{i,j}^k(x)u), \ k=1,2, $$ with its first eigenvalue $\lambda_k>0$ for $k=1,2,$ and in the context, $\lambda=:\min\{\lambda_1,\lambda_2\}.$ \\ We suppose the following conditions: \begin{itemize} \item[(H1)] $f,g: \Omega \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^N \mapsto \mathbb{R}$ are Caratheodory functions which satisfy \begin{gather*} |f(x,s,t,\xi,\eta)| \leq h_1(x,s,t)+k_1|\xi|^{\alpha_1}+k_2| \eta|^{\alpha_2}, \\ |g(x,s,t,\xi,\eta)| \leq h_2(x,s,t)+k_3|\xi|^{\alpha_3} +k_4|\eta|^{\alpha_4}, \end{gather*} where constant $\alpha_i,k_i \in \mathbb{R}_0^+$, $i = 1,2,3,4$; $h_1(x,s,t)$ and $h_2(x,s,t)$ are Caratheodory functions that satisfy the following conditions: \item[(H2)] for every $r>0$, $\sup _{|s| \leq r,\, |t| \leq r} h_i(\cdot ,s,t) \in L^p(\Omega)$, $\frac{2N}{N+1}
\overline u(x)\}$. Then $\Omega = A \cup B $. Obviously on $A$, $w^{+}=0$. In $B$, $Tu=\overline{u}$. Then the righthand side of \eqref{e2.12} is zero. That is, $$ \int_{\Omega}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega\cdot D_j\omega^+=0. $$ On $A$, $w^{+}=0$; on $B$, $\omega=\omega^+$. We can write the previous equation as $$ \int_{\Omega^+}\sum_{i,j=1}^N a_{ij}^1(x)D_i\omega^+\cdot D_j\omega^+=0. $$ Then according to the definition of the uniform elliptic operator, $$ \lambda{|D\omega^+|}^2\leq\int_{\Omega^+}\sum_{i,j=1}^N a_{ij}^1(x) D_i\omega^+\cdot D_j \omega^+=0. $$ Consequently, $w^+ = 0, x\in \Omega$. That is in $\Omega$, $u \le \overline u$. Similarly, we can prove that $\underline u \le u$ and $\underline v \le v\le \overline v$. From the definition of $T$, we know $Tu=u$ and $Tv=v$. Then by \eqref{e2.10}, we obtain that $(u,v) \in[W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)]^2$ is the solution of \eqref{e1.1}. The proof is completed. \end{proof} \subsection*{An example} In this section, we illustrate Theorem \ref{thm1.1}. \begin{equation} \label{e3.1} \begin{gathered} L_1 u=\lambda_1\phi_1(x)+\frac{2\lambda_1}{9}u+v+\lambda_1\phi_1|Du|^{ \frac{1}{2}} , \quad x \in \Omega, \\ L_2 v=\frac{3}{4}\lambda_2^2\phi_2(x)+\frac{\lambda_2^2}{12}u +\frac{\lambda_2}{4}v+ \frac{\sqrt{3}}{4}\lambda_2^{\frac{3}{2}}\phi_2(x)|Dv|^{\frac{1}{2}}, \quad x\in \Omega ,\\ u=v=0 , \quad x \in \partial \Omega. \end{gathered} \end{equation} Here $\Omega$ is a regular domain in ${\mathbb \mathbb{R}}^N (N>2)$ with smooth boundary $\partial \Omega$, and $$ \phi_i(x)=\frac{\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+\sup_{\Omega} |D\varphi_i|} \le 1. $$ In addition, $\lambda_i>0,\varphi_i(x)>0$ are the first eigenvalue and the corresponding eigenfunction of operator $L_i$ in $\Omega$ with zero-Dirichlet boundary value condition. Therefore, $$ L_i\phi_i(x)=\frac{L_i\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+ \sup_{\Omega}|D\varphi_i|} =\frac{\lambda_i\varphi_i(x)}{\sup_{\Omega}|\varphi_i|+\sup_{\Omega} |D\varphi_i|}=\lambda_i\phi_i(x). $$ When $2