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

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.