Existence theorems for fractional p-Laplacian problems

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalueperturbed problem depending on a real parameter λ under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weightedfractional p-Laplacian operator. Denoting by (λ_k)_k a sequence of eigenvalues obtainedvia mini–max methods and linking structures we prove the existence of (weak) solutionsboth when there exists k ∈ ℕ such that λ = λ_k and when λ /∉ (λ_k)_k. The paper isdivided 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 locallypositive or locally negative at 0 is taken into account. In the latter case, it is necessary toextend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina,Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var.9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linkingmethods.