On fractional Schr\"odinger equations with Hartree type nonlinearities

TL;DR

This paper investigates fractional Schr"odinger equations with Hartree nonlinearities, establishing existence, regularity, and decay properties of solutions for general nonlinearities of Berestycki-Lions type.

Contribution

It proves the existence of ground states and extends regularity and decay results for solutions of fractional Schr"odinger equations with Hartree-type nonlinearities.

Findings

Existence of ground state solutions established.

Regularity and decay properties of solutions extended.

Results apply to general nonlinearities of Berestycki-Lions type.

Abstract

Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} in the case of general nonlinearities $F \in C^1(\mathbb{R})$ of Berestycki-Lions type, when $N \geq 2$ and $\mu>0$ is fixed. Here $(-\Delta)^s$, $s \in (0,1)$, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $I_{\alpha}$, $\alpha \in (0,N)$. We prove existence of ground states of \eqref{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [25, 65].