On Dirichlet problems with singular nonlinearity of indefinite sign

TL;DR

This paper investigates the existence and nonexistence of positive solutions for a class of singular elliptic boundary value problems with indefinite sign, extending results to cases where the coefficient function changes sign.

Contribution

It provides new criteria for existence and nonexistence of solutions to singular problems with indefinite sign and variable coefficients.

Findings

Established conditions for existence of positive solutions.

Identified parameter ranges leading to nonexistence.

Extended results to sign-changing coefficient functions.

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $\alpha,\gamma>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-\Delta u=K\left( x\right) u^{-\alpha}-\lambda M\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\lambda>0$ is a real parameter. We mention that as a particular case our results apply to problems of the form $-\Delta u=m\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $m$ is allowed to change sign in $\Omega$.