On fractional Schrödinger equations with Hartree type nonlinearities

Abstract

Goal of this paper is to study the following doubly nonlocal equationin the case of general nonlinearities F ∈ C 1 (R) of Berestycki-Lions type, when N ≥ 2 and µ > 0 is fixed. Here (−∆) s , s ∈ (0, 1), denotes the fractional Laplacian, while the Hartreetype term is given by convolution with the Riesz potential I α , α ∈ (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 [25,65].