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

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 first the perturbation is (p−1)−sublinear near +∞, while in the second the perturbation is (p−1)−superlinear near +∞ and in the third we do not require asymptotic condition at +∞. Using variational methods together with truncation and comparison techniques, we show that for λ∈(0,λˆ1) - λ>0 is the parameter and λˆ1 being the principal eigenvalue of (−Δp,W1,p0(Ω)) - we have positive solutions, while for λ≥λˆ1, 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 +∞, provided that the perturbation is damped by a parameter