# Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation

## TL;DR

This paper investigates the inhomogeneous Gross-Pitaevskii equation, establishing criteria for solution existence and blow-up, classifying blow-up solutions, and analyzing the stability of standing waves using variational methods.

## Contribution

It provides a sharp threshold for global existence and blow-up, constructs minimal mass blow-up solutions, and studies the stability of standing waves.

## Key findings

- Sharp threshold for global existence and blow-up
- Classification of finite time blow-up solutions
- Existence and stability analysis of standing waves

## Abstract

We study the Cauchy problem for an inhomogeneous Gross-Pitaevskii equation. We first derive a sharp threshold for global existence and blow up of the solution. Then we construct and classify finite time blow up solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.