Problems involving the fractional $g$-Laplacian with Lack of Compactness

TL;DR

This paper establishes compact embeddings for radial functions in fractional Orlicz-Sobolev spaces and proves Lions-Lieb type results, aiding in the analysis of solutions to fractional g-Laplacian problems without the $ riangle_2$-condition.

Contribution

It introduces new compact embedding results and Lions-Lieb type theorems for fractional Orlicz-Sobolev spaces, facilitating the study of fractional g-Laplacian equations without the $ riangle_2$-condition.

Findings

Proved compact embedding of radial functions in fractional Orlicz-Sobolev spaces.

Established Lions and Lieb type concentration-compactness results.

Applied results to existence of solutions for fractional g-Laplacian problems.

Abstract

In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space $W^{s, G}\left(\mathbb{R}^{N}\right)$ consisting of radial functions, our target embedding spaces are of Orlicz type. Also, we prove a Lions and Lieb type results for $W^{s,G}\left(\mathbb{R}^{N}\right)$ that works together in a particular way to get a sequence whose the weak limit is nontrivial. As an application, we study the existence of solutions to Quasilinear elliptic problems in the whole space $\mathbb{R}^N$ involving the fractional $g-$Laplacian operator, where the conjugated function $\widetilde{G}$ of $G$ doesn't satisfy the $\Delta_2$-condition.