Existence of strictly positive solutions for sublinear elliptic problems in bounded domains

TL;DR

This paper investigates conditions under which strictly positive solutions exist for a class of sublinear elliptic boundary value problems involving sign-changing weights in bounded domains.

Contribution

It establishes new existence and nonexistence results for positive solutions of sublinear elliptic equations with discontinuous, sign-changing coefficients.

Findings

Existence of positive solutions under certain conditions on the weight function.

Nonexistence results when conditions are not met.

Extension of previous results to more general sign-changing weights.

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.