Ground state solutions for a class of Choquard equations with potential vanishing at infinity

This paper is dedicated to studying the following nonlinear Choquard equation−△u+V(x)u=(∫RNQ(y)F(u(y))|x−y|μdy)Q(x)f(u),u∈D1,2(RN), where N≥3, μ∈(0,N), V∈C(RN,[0,∞)), Q∈C(RN,(0,∞)), f∈C(R,R) and F(t)=∫0tf(s)ds. By combining the non-Nehari manifold approach with some new inequalities, we prove that the above equation has a ground state solution in the case when V(x)→0 as |x|→∞, where the strict monotonicity condition on f is not required. Moreover, by using perturbation method, we obtain the existence of a least energy solution in the zero mass case, i.e. V=0, where f satisfies the condition that F(t0)≠0 for some t0∈R instead of the usual Ambrosetti-Rabinowitz type condition. These results extend the ones in Alves, Figueiredo and Yang (2015) [1] and some related literature.