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

Abstract

<p style='text-indent:20px;'>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 <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\triangle)^{s}_{m}u+V(x)m(u) = f(x,u),\ x\in\mathbb{R}^{d}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ (-\triangle)^{s}_{m} $\end{document}</tex-math></inline-formula> is the fractional <inline-formula><tex-math id="M4">\begin{document}$ M $\end{document}</tex-math></inline-formula>-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.