preprint

Inserted: 31 may 2023

Year: 2023

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.