Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

We prove existence of infinitely many bound states solutions for the nonlinear Choquard equation in presence of the Riesz potential and an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: the constrained case (L2-norm of the solution is prescribed) and the unconstrained case. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions in [5], we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we nd a nonlocal counterpart of their multiplicity results. In particular we extend the existence results, due to Moroz and Van Schaftingen [TAMS, 2015].