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In this paper we study the following nonlinear fractional Hartree (orChoquard-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 ageneral subcritical nonlinearity. The goal is to prove existence of multiple(radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ oddor even.  We consider both the case $\mu>0$ fixed (and the mass $\int_{\mathbb{R}^N}u^2$ free) and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed (and thefrequency $\mu$ unknown). A key point in the proof is given by the research ofsuitable multidimensional odd paths, which was done in the local case byBerestycki and Lions [ARMA, 1983]. For equation \eqref{eq_abstract}, thenonlocalities play a special role in the construction of such paths. Inparticular, some properties of these paths are needed in the asymptotic study(as $\mu$ varies) of the mountain pass values of the unconstrained problem:this asymptotic behaviour is then exploited to describe the geometry of theconstrained problem and detect infinitely many normalized solutions for any$m>0$.  The found solutions satisfy in addition a Pohozaev identity: in this paper wefurther investigate the validity of this identity for solutions of doublynonlocal equations under a $C^1$-regularity.

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