Embedding Results and Fractional $p(x,\mathrm{.})$-Laplacian Problem in Unbounded Domains

In this paper, we prove a new continuous embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue spaces on unbounded domains. As an application, we study a class of nonlocal elliptic problems driven by the fractional $p(x,\cdot )$-Laplacian operator. Using variational methods combined with the established embedding result, we prove the existence of nontrivial weak solutions under suitable growth and regularity conditions on the nonlinearity.\ A significant analytical challenge addressed in this work arises from both the nonlocal behavior of the fractional $p(x,\cdot )$-Laplacian and the lack of compactness induced by the unboundedness of the domain.