Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

Abstract

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet p-Laplacian. We consider three cases. In the rst the perturbation is (p-1)-sublinear near $+\infty$, while in the second the perturbation is (p-1)-superlinear near $+infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in(0,\hat\lambda_1)$ ($\lambda> 0$ is the parameter and $\hat\lambda_1$ being the principal eigenvalue of $-\Delta_p,W_0^{1,p}(\Omega) $, we have positive solutions, while for $\lamba\geq \hat\lambda_1$, no positive solutions exist. In the \sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.