Topological structure of the solution set of singular equations with sign changing terms under Dirichlet boundary condition

Abstract

In this paper we establish existence of connected components of positive solutions of the equation $ -\Delta_{p} u = \lambda f(u)$ in $\Omega$, under Dirichlet boundary conditions, where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\Delta_{p}$ is the $p$-Laplacian, and $f \colon (0,\infty) \rightarrow \mathbb{R} $ is a continuous function which may blow up to $\pm \infty$ at the origin.