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

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 $\tilde{G}$ of $G$ doesn't satisfy the ${\Delta }_2$-condition.