Embedding and extension results in Fractional Musielak-Sobolev spaces

TL;DR

This paper explores the properties of fractional Musielak-Sobolev spaces, including inequalities, embeddings, and extension results, and applies variational methods to prove existence of solutions for nonlocal fractional problems.

Contribution

It introduces new fractional Musielak-Sobolev spaces, establishes key inequalities and embedding theorems, and demonstrates the existence of solutions to fractional boundary value problems.

Findings

Generalized Poincaré inequality established

Continuous and compact embeddings proved

Existence of nontrivial solutions demonstrated

Abstract

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the generalized Poincar\'e type inequality and some continuous and compact embedding theorems of these spaces. Moreover, we prove that any function in $W^sL_{\varPhi_{x,y}}(\Omega)$ may be extended to a function in $W^sL_{\varPhi_{x,y}}(\R^N)$, with $\Omega \subset \R^N$ is a bounded domain of class $C^{0,1}$. In addition, we establish a result relates to the complemented subspace in $W^s{L_{\varPhi_{x,y}}}\left( \R^N\right)$. As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.