Ground state solutions for Choquard equations with Hardy-Littlewood-Sobolev upper critical growth and potential vanishing at infinity

In this paper, we investigate the following non-autonomous Choquard equation -Delta u+V(x)u = (I-alpha *(vertical bar u vertical bar(2)(alpha)* + K(x)F(u))) 2(alpha)*vertical bar u vertical bar(2 alpha)* (-2)u + K(x)f(u)) in R-N, where N >= 3, I-alpha is the Riesz potential of order alpha is an element of (0, N), 2(alpha)* = N+alpha/N-2 is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality, the functions V, K is an element of C(R-N, R+) may vanish at infinity, and the function f is an element of C(R, R) is noncritical and F(t) = integral(t)(0) f (s)ds. Based on variational methods and the Hardy-type inequalities, using the mountain pass theorem and the Nehari manifold approach, we prove the existence of ground state solution. (C) 2019 Elsevier Inc. All rights reserved.