On Nonlinear Schrödinger Equations with Repulsive Inverse-Power Potentials

Abstract

In this paper, we consider the Cauchy problem for nonlinear Schrödinger equations with repulsive inverse-power potentials $$ i \partial _{t} u + \Delta u - c |x|^{-\sigma } u = \pm |u|^{\alpha }u, \quad c>0. $$ We study the local and global well-posedness, finite time blow-up and scattering in the energy space for the equation. These results extend a recent work of Miao-Zhang-Zheng (2018, arXiv:1809.06685) to a general class of inverse-power potentials and higher dimensions.