On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems

TL;DR

This paper introduces fractional anisotropic Musielak--Sobolev spaces to model complex nonlocal behaviors and applies them to establish existence results for nonlocal eigenvalue problems with variable growth.

Contribution

It develops a new class of fractional anisotropic Musielak--Sobolev spaces and applies them to nonlocal eigenvalue problems, extending existing models in PDE theory.

Findings

Established fundamental properties and embeddings of the new spaces

Proved existence of eigenvalues for nonlocal anisotropic problems

Unified several models in nonlocal PDE theory

Abstract

In this paper, we introduce and study a new class of fractional modular function spaces, called \emph{Fractional Anisotropic Musielak--Sobolev Spaces}, which generalize both the fractional Anisotropic Orlicz--Sobolev spaces and the Anisotropic fractional Sobolev spaces with variable exponent. These spaces are designed to handle anisotropic and heterogeneous behaviors that naturally arise in nonlocal and nonlinear models. We develop their fundamental properties and embedding results, establishing a solid variational framework. As an application, we investigate a class of nonlocal anisotropic eigenvalue problems involving variable growth and direction-dependent fractional integro-differential operators. We prove the existence of eigenvalues by means of critical point theory and modular analysis. Our results extend and unify several existing models in the theory of nonlocal partial differential equations.