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.