On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems
Mohammed Srati
TL;DR
The paper develops a novel framework of Fractional Anisotropic Musielak-Sobolev spaces to model nonlocal, direction-dependent diffusion with nonstandard growth. It builds a rigorous variational setting using modular energies and directional fractional operators, proving embedding and compactness results that support critical point methods. Two eigenvalue existence results for a nonlocal anisotropic problem $(P_a)$ are established under different growth assumptions on the variable exponent $q(x)$, employing Mountain Pass and Ekeland variational principles. By unifying fractional anisotropic Orlicz–Sobolev and anisotropic fractional Sobolev with variable exponent frameworks, the work provides robust tools for nonlocal PDEs in heterogeneous media with potential applications in physics and engineering.
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.