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