*Published Paper*

**Inserted:** 31 may 2023

**Last Updated:** 15 may 2024

**Journal:** Advanced Nonlinear Studies

**Volume:** 24

**Number:** 2

**Pages:** 303-334

**Year:** 2024

**Doi:** https://doi.org/10.1515/ans-2023-0110

**Abstract:**

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_{\alphaF}**(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 mass $\int_{\mathbb{R}^N}
u^2$ free) and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed (and the
frequency $\mu$ unknown). 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 ARMA, 1983. For equation \eqref{eq_{abstract},} the
nonlocalities play a special role in the construction of such paths. 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:
this asymptotic behaviour is 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.**

**Keywords:**
fractional Laplacian, Pohozaev identity, infinitely many solutions, Double nonlocality, nonlinear Choquard Pekar equation, normalized solutions