  This paper aims to establish the existence of a weak solution for the
following problem: \begin{equation*} (-\Delta)^{s}_{\mathcal{H}}u(x)
+V(x)h(x,x,|u|)u(x)=\left(\int_{\mathbb{R}^{N}}\dfrac{K(y)F(u(y))}{|x-y|^\lambda}dy
\right) K(x)f(u(x)) \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} where $N\geq
1$, $s\in(0,1), \lambda\in(0,N), \mathcal{H}(x,y,t)=\int_{0}^{|t|} h(x,y,r)r\
dr,$ $ h:\mathbb{R}^{N}\times\mathbb{R}^{N}\times
[0,\infty)\rightarrow[0,\infty)$ is a generalized $N$-function and
$(-\Delta)^{s}_{\mathcal{H}}$ is a generalized fractional Laplace operator. The
functions $V,K:\mathbb{R}^{N}\rightarrow (0,\infty)$, non-linear function
$f:\mathbb{R}\rightarrow \mathbb{R}$ are continuous and $
F(t)=\int_{0}^{t}f(r)dr.$ First, we introduce the homogeneous fractional
Musielak-Sobolev space and investigate their properties. After that, we pose
the given problem in that space. To establish our existence results, we prove
and use the suitable version of Hardy-Littlewood-Sobolev inequality for
Lebesque Musielak spaces together with variational technique based on the
mountain pass theorem. We also prove the existence of a ground state solution
by the method of Nehari manifold.
