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

TL;DR

This paper introduces a new anisotropic fractional Sobolev space with variable exponent, establishes its fundamental properties, and applies it to prove the existence of solutions for nonlocal problems using variational methods.

Contribution

It develops a novel fractional anisotropic Sobolev space with variable exponent and analyzes the associated anisotropic fractional p-Laplacian operator, including embedding and functional properties.

Findings

Established completeness, separability, and reflexivity of the new space.

Proved continuous and compact embedding results.

Proved existence of weak solutions for nonlocal problems using variational principles.

Abstract

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 $\vec{p}(.,.)$-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.