On the Structure of the Solution Set of a Sign Changing Perturbation of the p-Laplacian under Dirichlet Boundary Condition

Abstract

In a recent paper D. D. Hai showed that the equation $ -\Delta_{p} u = \lambda f(u) \mbox{in} \Omega$, under Dirichlet boundary condition, where $\Omega \subset {\bf R^N}$ is a bounded domain with smooth boundary $\partial\Omega$, $\Delta_{p}$ is the p-Laplacian, $f : (0,\infty) \rightarrow {\bf R} $ is a continuous function which may blow up to $\pm \infty$ at the origin, admits a solution if $\lambda > \lambda_0$ and has no solution if $0 < \lambda < \lambda_0$. In this paper we show that the solution set $\mathcal{S}$ of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.