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

Abstract

In this paper we study the following nonlinear fractional Choquard-Pekar equation \begin{equation}\label{eq_abstract} (-Δ)^s u + μu =(I_α*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$\ast$} \end{equation} where $\mu >0$, $s\in (0,1)$, $N\ge 2$, $\alpha \in (0,N)$, $I_{\alpha }\sim \frac{1}{|x{|}^{N-\alpha }}$ 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 $s=1$ in [Calc. Var. PDEs, 2022]. 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 \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.

Advanced Nonlinear Studies (in press)

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Cited by in corpus (1)

• Asymptotic decay of solutions for sublinear fractional Choquard equations