Positive solutions of indefinite semipositone problems via sub-super solutions

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 1$, be a smooth bounded domain, and let $m:\Omega \to \mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem $-\Delta u=\lambda m(x)(f(u)-k)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\lambda ,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $\lambda$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

To appear in Differential and Integral Equations