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

## TL;DR

This paper investigates the nonlinear Schrödinger equation with attractive inverse-power potentials, establishing conditions for global solutions, blow-up, and the existence of energy minimizers, especially near critical mass thresholds.

## Contribution

It provides a sharp threshold for well-posedness and blow-up, and analyzes minimizers and their blow-up behavior in critical cases, advancing understanding of these equations with inverse-power potentials.

## Key findings

- Identified a sharp threshold for global well-posedness and blow-up.
- Proved existence and non-existence of energy minimizers under mass constraints.
- Analyzed blow-up behavior of minimizers near critical mass values.

## Abstract

We study the Cauchy problem for nonlinear Schr\"odinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and non-existence of minimizers for the energy functional with prescribed mass constraint. In the mass-critical case, we also study the blow-up behavior of minimizers when the mass tends to a critical value.