Electron. J. Diff. Equ., Vol. 2011 (2011), No. 164, pp. 1-8.

Existence of positive solutions for semilinear elliptic systems with indefinite weight

Ruipeng Chen

Abstract:
This article concerns the existence of positive solutions of semilinear elliptic system
$$\displaylines{ -\Delta u=\lambda a(x)f(v),\quad\hbox{in }\Omega,\cr -\Delta v=\lambda b(x)g(u),\quad\hbox{in }\Omega,\cr u=0=v,\quad \hbox{on } \partial\Omega, }$$
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.

Submitted September 13, 2011. Published December 13, 2011.
Math Subject Classifications: 35J45.
Key Words: Semilinear elliptic systems; indefinite weight; positive solutions; existence of solutions.

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Ruipeng Chen
Department of Mathematics
Northwest Normal University
Lanzhou, 730070, China
email: ruipengchen@126.com

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