Remarks on the fractional inhomogeneous Hartree equation

Abstract

This paper studies the inhomogeneous fractional Schödinger equation $$i\dot{u}-(-\Delta {)}^{s}u=\pm (I_{\alpha }\ast |\cdot {|}^b|u{|}^p)|x{|}^b|u{|}^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.