Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

TL;DR

This paper proves the existence of solutions for a nonlocal fractional elliptic problem within fractional Orlicz-Sobolev spaces, extending classical Sobolev spaces and establishing key functional properties and embeddings.

Contribution

It introduces a generalized fractional Sobolev space $W^sL_A$, analyzes its properties, and applies these to demonstrate solution existence for nonlocal elliptic problems.

Findings

Extended fractional Sobolev spaces to include Orlicz functions

Proved completeness, reflexivity, and separability of $W^sL_A$

Established continuous and compact embeddings into Lebesgue spaces

Abstract

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $W^{s,p}$ to include the general case $W^sL_A$, where $A$ is an N-function and $s\in (0,1)$. We are concerned with some qualitative properties of the space $W^sL_A$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.