On fractional Schrödinger equations with Hartree type nonlinearities

Abstract

<abstract><p>Goal of this paper is to study the following doubly nonlocal equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \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} </tex-math> </disp-formula></p> <p>in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu &gt; 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 ^{[<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b61">61</xref>]}.</p></abstract>