Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

TL;DR

This paper establishes new embedding theorems for fractional Orlicz-Sobolev spaces and applies them to prove the existence of multiple solutions for non-local fractional Schrödinger equations involving the fractional M-Laplace operator.

Contribution

It introduces novel continuous and compact embedding theorems for fractional Orlicz-Sobolev spaces and uses them to demonstrate multiple solutions for related non-local equations.

Findings

New embedding theorems for fractional Orlicz-Sobolev spaces

Existence of infinitely many solutions for fractional Schrödinger equations

Application of Fountain theorem to non-local problems

Abstract

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schr\"{o}dinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}u+V(x)m(u)=f(x,u),\ x\in\mathbb{R}^{d},$$ where $0<s<1$, $d\geq2$ and $(-\triangle)^{s}_{m}$ is the fractional $M$-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.