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

Abstract We prove existence of infinitely many solutions $$u \in H^1_r({\mathbb {R}}^N)$$ u ∈ H r 1 ( R N ) for the nonlinear Choquard equation $$\begin{aligned} - {\varDelta } u + \mu =(I_\alpha *F(u)) f(u) \quad \hbox {in}\ {\mathbb {R}}^N, \end{aligned}$$ - Δ + μ = I α ∗ F f in , where $$N\ge 3$$ ≥ 3 , $$\alpha (0,N)$$ 0 $$I_\alpha (x) := \frac{{\varGamma }(\frac{N-\alpha }{2})}{{\varGamma }(\frac{\alpha }{2}) \pi ^{N/2} 2^\alpha \frac{1}{|x|^{N- \alpha }}$$ x : Γ 2 π / | $$x {R}}^N \setminus \{0\}$$ \ { } is Riesz potential, and F an almost optimal subcritical nonlinearity, assumed odd or even. analyze two cases: $$\mu $$ a fixed positive constant unknown $$L^2$$ L -norm solution prescribed, i.e. $$\int _{{\mathbb {R}}^N} |u|^2 =m>0$$ ∫ m > . Since presence nonlocality prevents to apply classical approach, introduced by Berestycki Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement new construction multidimensional paths, some estimates potential play essential role, find nonlocal counterpart their multiplicity results. In particular extend results due Moroz Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).