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

We prove existence of infinitely many solutions u∈Hr1(RN) for the nonlinear Choquard equation -Δu+μu=(Iα∗F(u))f(u)inRN,where N≥ 3 , α∈ (0 , N) , Iα(x):=Γ(N-α2)Γ(α2)πN/22α1|x|N-α, x∈ R^N\ { 0 } is the Riesz potential, and F is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: μ is a fixed positive constant or μ is unknown and the L²-norm of the solution is prescribed, i.e. ∫RN|u|2=m>0. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results due to Moroz and Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).