On doubly nonlocal fractional elliptic equations

TL;DR

This paper investigates the existence of solutions to nonlocal fractional elliptic equations involving the fractional Laplacian, employing variational methods and the Mountain Pass Theorem under general assumptions.

Contribution

It provides a new fractional version of a classical theorem for Laplacian equations, establishing solution existence with minimal assumptions.

Findings

Existence of nontrivial solutions proven using variational methods.

Analysis of fractional spaces necessary for nonlinear methods.

Results are novel in the context of doubly nonlocal fractional equations.

Abstract

This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.