On nonlinear Schrödinger equations with repulsive inverse-power potentials

Abstract

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equations with repulsive inverse-power potentials \[ i \partial_t u + Δu - c |x|^{-σ} u = \pm |u|^αu, \quad c>0. \] 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 Schrödinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.

arXiv comment38 pages, 1 figure