Ground state solution for a generalized Choquard Schrodinger equation with vanishing potential in homogeneous fractional Musielak Sobolev spaces

Abstract

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.