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

## TL;DR

This paper investigates the well-posedness, blow-up, and scattering of nonlinear Schrödinger equations with repulsive inverse-power potentials, extending previous results to broader potentials and higher dimensions in the energy space.

## Contribution

It extends existing work on Coulomb potentials to a wider class of inverse-power potentials and higher-dimensional cases for nonlinear Schrödinger equations.

## Key findings

- Established local and global well-posedness in energy space.
- Analyzed finite time blow-up conditions.
- Proved scattering results for the equations.

## Abstract

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.