Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians - Uriel Kaufmann; Julio Rossi; Raul Vidal (Electronic Journal of Qualitative Theory of Differential Equations, 2017)
In this article we extend the Sobolev spaces with variable exponents to include the fractional case, and we prove a compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As an application we prove the existence and uniqueness of a solution for a nonlocal problem involving the fractional $p(x)$-Laplacian.