On fractional Schrödinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation\r\n\begin{equation}\label{eq_abstract}\r\n(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P}\r\n\end{equation}\r\nin the case of general nonlinearities $F \in C^1(\R)$ 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 $I_{\alpha}$, $\alpha \in (0,N)$. \r\nWe prove existence of ground states of \eqref{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in \cite{DSS1, MS2}.