Compact embedding theorems and a Lions' type Lemma for fractional Orlicz-Sobolev spaces

TL;DR

This paper establishes compact embedding theorems and a Lions' type lemma for fractional Orlicz-Sobolev spaces, and applies these results to prove existence of solutions for nonlinear Schrödinger equations.

Contribution

It introduces new compactness and vanishing lemmas for fractional Orlicz-Sobolev spaces and applies them to nonlinear PDEs, extending classical results to more general function spaces.

Findings

Proved compact embedding of weighted fractional Orlicz-Sobolev spaces into Orlicz spaces.

Established a Lions' type vanishing lemma for fractional Orlicz-Sobolev spaces.

Demonstrated existence of ground state solutions for nonlinear Schrödinger equations using variational methods.

Abstract

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schr\"{o}dinger equations, taking into account unbounded or bounded potentials.