On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

TL;DR

This paper establishes new embedding theorems for fractional Musielak-Sobolev spaces in unbounded domains and applies these results to prove the existence of solutions for a class of nonlinear Schrödinger equations.

Contribution

It introduces the first continuous and compact embedding results for fractional Musielak-Sobolev spaces in bounded domains and applies them to PDEs using variational methods.

Findings

Established new embedding theorems for fractional Musielak-Sobolev spaces in bounded domains.

Proved existence of nontrivial weak solutions for nonlinear Schrödinger equations with variable exponents.

Connected the theory of fractional Musielak-Sobolev spaces with PDE applications.

Abstract

This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in $\mathbb{R}^d$. As an application, using the variational methods, we obtain the existence of nontrivial weak solution for the following Schr\"odinger equation $$ (-\Delta)_{g_{x,y}}^s u+V(x)g(x,x,u)=b(x)\vert u\vert^{p(x)-2}u,\ \text{for all}\ x\in \mathbb{R}^d,$$ where $(-\Delta)_{g_{x,y}}^s$ is the fractional Museilak $g_{x,y}$-Laplacian, $V$ is a potential function, $b\in L^{\delta^{'}(x)}(\mathbb{R}^d)$, and $p,\delta\in C\left(\mathbb{R}^d,(1,+\infty)\right)\cap L^{\infty}(\mathbb{R}^d)$. We would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers in this topic, see \cite{M1,M8,M9}. Note that, all these latter works dealt with bounded case and there are no results devoted for the fractional Musielak-Sobolev spaces in $\mathbb{R}^d$. Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's.