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

Abstract

In this paper, we investigate the following non-autonomous Choquard equation−Δu+V(x)u=(Iα⁎(|u|2α⁎+K(x)F(u)))(2α⁎|u|2α⁎−2u+K(x)f(u))inRN, where N⩾3, Iα is the Riesz potential of order α∈(0,N), 2α⁎=N+αN−2 is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality, the functions V,K∈C(RN,R+) may vanish at infinity, and the function f∈C(R,R) is noncritical and F(t)=∫0tf(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.