## Differential and Integral Equations

### Instability of standing waves for nonlinear Schrödinger equations with potentials

#### Abstract

We study the instability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with an attractive power nonlinearity $|u|^{p-1}u$ and a potential $V(x)$ in ${\mathbb{R}}^n$. Here, $\omega>0$ and $\phi_{\omega}(x)$ is a minimal action solution of the stationary problem. Under suitable assumptions on $V(x)$, we show that if $p>1+4/n$, $e^{i \omega t}\phi_{\omega}(x)$ is unstable for sufficiently large $\omega$. For example, our theorem covers a harmonic potential $V(x)=|x|^2$, to which the arguments in the previous papers [2], [14], and [19] are not directly applicable. As another application, we also prove a similar result for a nonlinear Schrödinger equation with a constant magnetic field.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 6 (2003), 691-706.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060607

Mathematical Reviews number (MathSciNet)
MR1973275

Zentralblatt MATH identifier
1031.35131

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35R25: Improperly posed problems

#### Citation

Fukuizumi, Reika; Ohta, Masahito. Instability of standing waves for nonlinear Schrödinger equations with potentials. Differential Integral Equations 16 (2003), no. 6, 691--706. https://projecteuclid.org/euclid.die/1356060607