# Infinitely many solutions for a class of fractional Orlicz-Sobolev Schr\"odinger equations

## TL;DR

This paper proves the existence of infinitely many solutions for a class of fractional Orlicz-Sobolev Schrödinger equations using variational methods and a new compact embedding theorem.

## Contribution

It introduces a new compact embedding theorem for fractional Orlicz-Sobolev spaces and applies it to establish multiple solutions for the Schrödinger equations.

## Key findings

- Established a new compact embedding theorem.
- Proved existence of infinitely many solutions.
- Applied variational methods with the Fountain theorem.

## Abstract

In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schr\"odinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}+V(x)m(u)u=f(x,u),\ x\in\mathbb{R}^{N},$$ where $s\in ]0,1[$, $N\geq2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u| \rightarrow\infty$. The proof is based on the variant Fountain theorem established by Zou.