Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 164, pp. 1-8.
Title: Existence of positive solutions for semilinear elliptic
systems with indefinite weight
Author: Ruipeng Chen (Northwest Normal Univ., Lanzhou, China)
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.