*Published Paper*

**Inserted:** 10 may 2022

**Last Updated:** 31 may 2023

**Journal:** Mathematics in Engineering

**Volume:** 4

**Number:** 6

**Pages:** 1-33

**Year:** 2021

**Doi:** 10.3934/mine.2022056

**Abstract:**

Goal of this paper is to study the following doubly nonlocal equation
\begin{equation}\label{eq_{abstract}} (- \Delta)^{s} u + \mu u =
(I_{\alphaF}**(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.
**

**Keywords:**
regularity, fractional Laplacian, Asymptotic decay, symmetric solutions, nonlinear Schrödinger equation, Double nonlocality, Choquard nonlinearity, Hartree term