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

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 Schrödinger equation $$(-\mathrm{\Delta }{)}_{g_{x,y}}^{s}u+V(x)g(x,x,u)=b(x)\mathrm{\mid }u{\mathrm{\mid }}^{p(x)-2}u,\text{ for all }x\in {\mathbb{R}}^d,$$ where $(-\mathrm{\Delta }{)}_{g_{x,y}}^s$ is the fractional Museilak $g_{x,y}$-Laplacian, $V$ is a potential function, $b\in L^{{\delta }^{{}^{\mathrm{\prime }}}(x)}({\mathbb{R}}^d)$, and $p,\delta \in C({\mathbb{R}}^d,(1,+\mathrm{\infty }))\cap L^{\mathrm{\infty }}({\mathbb{R}}^d)$. 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.