On fractional Schrödinger equations with Hartree type nonlinearities

Goal of this paper is to study the following doubly nonlocal equation \begin{document}$(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}})$ \end{document} 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 (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [ 23 , 61 ] .