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

We study the existence of positive solutions for perturbations of theclassical 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 theperturbation is (p-1)-superlinear near $+infty$ and in the third we do not requireasymptotic condition at $+\infty$. Using variational methods together with truncationand 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 positivesolution is unique under a suitable monotonicity condition, while in the "superlinearcase" we produce the existence of a smallest positive solution. Finally, we point outan existence result of a positive solution without requiring asymptotic condition at$+\infty$, provided that the perturbation is damped by a parameter.