Existence theorems for fractional p-Laplacian problems

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional p-Laplacian operator. Using a sequence of eigenvalues obtained via mini-max methods and linking structures we prove the existence of (weak) solutions both in the resonance and in the non resonance case The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at 0 is taken into account. In the latter case it is necessary to extend the main results reported in a recent paper of Iannizzotto, Liu, Perera and Squassina. In both cases, the existence of solutions is achieved via linking methods.