Strictly positive solutions for one-dimensional nonlinear problems involving the p-Laplacian

TL;DR

This paper investigates the existence of strictly positive solutions for one-dimensional nonlinear elliptic problems involving the p-Laplacian operator, with new results even when the potential term c is zero.

Contribution

The study provides novel existence results for positive solutions to p-Laplacian problems with sign-changing weights and potential terms, extending previous work.

Findings

Established existence of positive solutions under broad conditions.

Extended results to cases with zero potential term c.

Provided new insights even when c is identically zero.

Abstract

Let $\Omega$ be a bounded open interval, and let $p>1$ and $q\in\left(0,p-1\right) $. Let $m\in L^{p^{\prime}}\left(\Omega\right) $ and $0\leq c\in L^{\infty}\left(\Omega\right) $. We study existence of strictly positive solutions for elliptic problems of the form $-\left(\left\| u^{\prime}\right\|^{p-2}u^{\prime}\right) ^{\prime}+c\left(x\right) u^{p-1}=m\left(x\right) u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$. We mention that our results are new even in the case $c\equiv0$.