Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation \begin{equation}\label{eq_abstract} (-\Delta)^s u + \mu u =(I_\alpha*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$*$} \end{equation} where $\mu>0$, $s \in (0,1)$, $N \geq 2$, $\alpha \in (0,N)$, $I_\alpha \sim \frac{1}{|x|^{N-\alpha}}$ is the Riesz potential, and $F$ is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ odd or even: we consider both the case $\mu>0$ fixed and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed. Here we also simplify some arguments developed for $s=1$ \cite{CGT4}. A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions \cite{BL2}; for \eqref{eq_abstract} the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as $\mu$ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any $m>0$. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a $C^1$-regularity.