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

We prove existence of infinitely many solutions $u \in H^1_r(\mathbb{R}^N)$ for the nonlinear Choquard equation $$ - \Delta u + \mu u =(I_\alpha*F(u)) f(u) \quad \hbox{in}\ \mathbb{R}^N, $$ where $N\geq 3$, $\alpha\in (0,N)$, $I_\alpha(x) := \frac{\Gamma(\frac{N-\alpha}{2})}{\Gamma(\frac{\alpha}{2}) \pi^{N/2} 2^\alpha } \frac{1}{|x|^{N- \alpha}}$, $x \in \mathbb{R}^N \setminus\{0\}$ is the Riesz potential, and $F$ is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: $\mu$ is a fixed positive constant or $\mu$ is unknown and the $L^2$-norm of the solution is prescribed, i.e. $\int_{\mathbb{R}^N} |u|^2 =m>0$. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions in \cite{BL2}, 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 in \cite{MS2}, due to Moroz and Van Schaftingen.