Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem

We consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. We assume that the perturbation is (p(z)−1)-sublinear, and we prove an existence and nonexistence theorem for positive solutions as the parameter λ moves on the positive semiaxis. We also show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the minimal solution map.