Existence theorems for fractional p-Laplacian problems

Abstract

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. 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 [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.