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

This paper aims to establish the existence of a weak solution for the following problem: (-Delta)Hsu(x)+V(x)h(x,x,|u|)u(x)=integral RNK(y)F(u(y))|x-y|lambda dyK(x)f(u(x)), in RN where N >= 1, s is an element of(0,1),lambda is an element of(0,N),H(x,y,t)=integral 0|t|h(x,y,r)rdr,h:RNxRNx[0,infinity)->[0,infinity) is a generalized N-function and (-Delta)Hs is a generalized fractional Laplace operator. The functions V,K:RN ->(0,infinity), non-linear function f:R -> R are continuous and F(t)=integral 0tf(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 use variational technique based on the mountain pass theorem. We also prove the existence of a ground state solution by the method of Nehari manifold.