On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent

The content of this paper is at the interplay between function spaces Lᵖ⁽ˣ⁾ and Wk, p(x) with variable exponents and fractional Sobolev spaces Ws, p. We are concerned with some qualitative properties of the fractional Sobolev space Ws, q(x), p(x, y), where q and p are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.