On nonlinear Schr\"odinger equations with repulsive inverse-power potentials

In this paper, we consider the Cauchy problem for the nonlinear Schr\"odinger 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 $H^1$ for the equation. These results extend a recent work of Miao-Zhang-Zheng [Nonlinear Schr\"odinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.