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

Abstract

Let $\Omega$ be a bounded open interval, let $p>1$ and $\gamma >0$, and let $m:\Omega \to \mathbb{R}$ 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 $-({\left|u^{\prime }\right|}^{p-2}{u}^{\prime }{)}^{\prime }=m\left(x\right)u^{-\gamma }$ in $\Omega$, $u=0$ on $\partial \Omega$. As a consequence we also derive existence results for other related nonlinearities.

To appear in Advances in Nonlinear Analysis