Existence of positive solutions of a superlinear boundary value problem with indefinite weight

Abstract

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u^{\prime \prime }+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g:[0,+\infty [\to \mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^p$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.

12 pages, 4 PNG figures