One-dimensional singular problems involving the p-Laplacian and nonlinearities indefinite in sign

TL;DR

This paper investigates the existence of positive solutions for one-dimensional singular p-Laplacian problems with sign-changing nonlinearities, providing conditions for existence and nonexistence in bounded intervals.

Contribution

It offers new criteria for the existence and nonexistence of positive solutions to singular p-Laplacian problems with indefinite nonlinearities in one dimension.

Findings

Established conditions for positive solution existence.

Derived nonexistence results under certain conditions.

Extended results to related nonlinear problems.

Abstract

Let $\Omega$ be a bounded open interval, let $p>1$ and $\gamma>0$, and let $m:\Omega\rightarrow\mathbb{R}$ be a function that may change sign in $\Omega $. In this article we study the existence and nonexistence of positive solutions for one-dimensional singular problems of the form $-(\left\vert u^{\prime}\right\vert ^{p-2}u^{\prime})^{\prime}=m\left( x\right) u^{-\gamma}$ in $\Omega$, $u=0$ on $\partial\Omega$. As a consequence we also derive existence results for other related nonlinearities.