Infinitely many bound states for Choquard equations with local nonlinearities

Abstract

In this paper, we consider the following Choquard equation (CH)−Δu+u=(Iα∗|u|p)|u|p−2u+V(x)|u|q−2uinRN,u∈H1(RN),where N≥3, α∈((N−4)+,N), p∈[2,N+αN−2), q∈(2,2NN−2)∩(1+(p−1)(N−2)αN2,1+2N(N−α)+N2(p−1)(N−2)α) and Iα is the Riesz potential. Under some suitable decay assumptions but without any symmetry property on V(x), we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large.