Anisotropic fractional Sobolev spaces with variable exponent and application to nonlocal problems

The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, …) are established, including the continuous and compact embedding results. Moreover, some functional proprieties of anisotropic fractional $\vec{p}(\mathrm{.},\mathrm{.})$-Laplacian operator are proved. As an application, we use the mountain pass theorem and Ekeland’s variational principle to ensure the existence of a weak solution for a nonlocal anisotropic problem with variable exponent.