Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians

Abstract

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.

Keywords

• variable exponents
• sobolev spaces
• fractional laplacian