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 $ -Δ_{p} u = λf(u) \mbox{in} Ω$, under Dirichlet boundary condition, where $\Omega \subset {\mathbf{R}}^{\mathbf{N}}$ is a bounded domain with smooth boundary $\partial \Omega$, ${\Delta }_p$ is the p-Laplacian, $f:(0,\infty )\to \mathbf{R}$ 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.

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