# Strong instability of standing waves for nonlinear Schr\"{o}dinger equations with a partial confinement

## TL;DR

This paper proves that ground state solutions of certain nonlinear Schrödinger equations with partial confinement are strongly unstable and blow up under specific critical or supercritical conditions.

## Contribution

It establishes the strong instability of ground states for NLS with a harmonic potential in higher dimensions when the nonlinearity is critical or supercritical.

## Key findings

- Ground states are strongly unstable by blowup under specified conditions.
- Instability occurs for nonlinearities that are L^2-critical or supercritical in N-1 dimensions.
- Results apply to N ≥ 2 with a one-dimensional harmonic potential.

## Abstract

We study the instability of standing wave solutions for nonlinear Schr\"{o}dinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.