On fractional Schrödinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation

$$(-\mathrm{\Delta }{)}^{s}u+\mu u=(I_{\alpha }\ast F(u))F^{\prime }(u)\mathrm{i}\mathrm{n}{\mathbb{R}}^N(\mathrm{P})$$

in the case of general nonlinearities $F\in C^1(\mathbb{R})$ of Berestycki-Lions type, when $N\ge 2$ and $\mu >0$ is fixed. Here $(-\mathrm{\Delta }{)}^s$, $s\in (0,1)$, 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 (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in ^[23,61].