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 Schrödinger 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.
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