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: { − Δ u + V ( x ) u = ( W ⁎ F ( x , u ) ) f ( x , u ) , x ∈ R N ( N ≥ 2 ) , u ∈ H 1 ( R N ) , where V ∈ C ( R N , 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.