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 λ under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional p-Laplacian operator. Denoting by (λ k ) k a sequence of eigenvalues obtained via mini-max methods and linking structures we prove the existence of (weak) solutions both when there exists k ∈ N such that λ = λ k and when λ / ∈ (λ k ) k . 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. 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 linking methods.

Fractional p‐Laplacian problem with indefinite weight in RN: Eigenvalues and existence

“…Concerning the fractional ‐Laplacian eigenvalue problem defined in bounded domain, which has been presented much interesting results, see, for example, previous studies 5–14 and the references therein. In particular, for the following eigenvalue problem where , , is a bounded domain.…”

“…They deduced the conclusion (A) of the problem () and even the associated eigenfunction to can be positive or negative. When , Piersanti and Pucci 12 dealt with the problem () and verified that the problem exists a sequence of eigenvalues which converges to infinity via minimax methods.…”

“…On the existence of solution for fractional ‐Laplacian elliptic problem, there have been many results obtained by researchers for positive weight case; see, eg, previous studies 8,11,12,18–20 and the references therein. In particular, Piersanti and Pucci 12 considered the existence of solution to the following elliptic problem involving the fractional ‐Laplacian in bounded domain where is a positive weight, is a Carathéodory function. Xiang et al 20 investigated the existence of solution for the following problem where is a real parameter, …”

A Brezis-Nirenberg type result for a nonlocal fractional operator

“…See, for instance, the papers of Di Castro, Kuusi, and Palatucci , Franzina and Palatucci , Kuusi, Mingione, and Sire , Lindgren and Lindqvist , and the famous work of Caffarelli for the motivations that led to its introduction. See also the papers of Pucci and her co‐authors , where some existence and multiplicity results for fractional problems involving the ‐Laplacian operator are obtained using variational methods. Namely, for , and smooth enough, the fractional ‐Laplacian operator is defined as where is the Euclidean ball centered at with radius .…”

Nonlinear equations involving the square root of the Laplacian

“…Moreover, a lot of interest has been devoted to elliptic equations involving the fractions of the Laplacian, (see, among others, the papers [1,2,3,5,8,14,24,28,35,40] as well as [7,25,27,30,31,32,34] and the references therein). See also the papers [4,37] for related topics.…”