Remarks on the fractional inhomogeneous Hartree equation

This paper studies the inhomogeneous fractional Sch"odinger equation $$i\dot{u}-(-\mathrm{\Delta }{)}^{s}u=\pm (I_{\alpha }\ast \mathrm{\mid }\cdot {\mathrm{\mid }}^b\mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }x{\mathrm{\mid }}^b\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\mathrm{.}$$ 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.