Singular and Superlinear Perturbations of the Eigenvalue Problem for the Dirichlet p-Laplacian

Abstract

We consider a nonlinear Dirichlet problem, driven by the p-Laplacian with a reaction involving two parameters $\lambda \in {\mathbb {R}}, \theta >0$. We view the problem as a perturbation of the classical eigenvalue problem for the Dirichlet problem. The perturbation consists of a parametric singular term and of a superlinear term. We prove a nonexistence and a multiplicity results in terms of the principal eigenvalue ${\hat{\lambda }}_1>0$ of $(-\Delta _p, W_0^{1,p}(\Omega ))$. So, we show that if $\lambda \ge {\hat{\lambda }}_1$ and $\theta >0$, then the problem has no positive solution, while if $\lambda <{\hat{\lambda }}_1$ and $\theta >0$ is suitably small (depending on $\lambda$), there are two positive smooth solutions.