Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity

Abstract

This paper focuses on the study of ground states and multiple solutions for the following non-autonomous Choquard-Pekar equation:{−Δu+V(x)u=(W⁎F(x,u))f(x,u),x∈RN(N≥2),u∈H1(RN), where V∈C(RN,R). We consider first the case V changes sign which turns the problem into a indefinite case, and obtain the existence of nontrivial solution and infinitely many distinct pairs of solutions under a local super-linear condition assumed on the nonlinearity. For the case V is 1-periodic and positive, ground state solution and infinitely many solutions are established further by using the generalized Nehari manifold method. We finally give some non-existence criteria via a generalized Pohožaev identity established for the general potentials V and W.