Positive solutions of indefinite semipositone problems via sub-super solutions

TL;DR

This paper establishes the existence of positive solutions for indefinite semipositone problems involving the Laplacian, using sub-supersolutions and rescaling techniques, especially when the negative part of the weight function is small.

Contribution

It introduces new existence results for positive solutions of semipositone problems with sign-changing weights, extending previous methods with a focus on sub-supersolutions and rescaling.

Findings

Positive solutions exist for certain parameter ranges.

Small negative parts of the weight function are crucial.

Method applies to sublinear and singular nonlinearities.

Abstract

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ 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.