General fractional Sobolev space with variable exponent and applications to nonlocal problems

In this paper, we extend the fractional Sobolev spaces with variable exponents W-s,W-p(x,W-y) to include the general fractional case W-K(s,p(x,y)), where p is a variable exponent, s is an element of (0, 1) and K is a suitable kernel. We are concerned with some qualitative properties of the space W-K(s,p(x,y)) (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and a compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal p(x, .)-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type L-K(p(x,.)).