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.
Categories: math.AP