Existence and multiplicity results for the fractional Laplacian in bounded domains

TL;DR

This paper investigates the existence and multiplicity of solutions for fractional Laplacian problems in bounded domains, using topological methods and critical point theory to establish new results.

Contribution

It provides new existence and multiplicity results for fractional Laplacian problems, especially near eigenvalues, employing topological and variational techniques.

Findings

Existence results for perturbed fractional Laplacian problems.

Multiplicity of solutions near eigenvalues.

Application of critical point theorem of Marino and Saccon.

Abstract

In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.