Instability of standing waves for a quasi-linear Schrödinger equation in the critical case

Xiaoguang Li; Chaohe Zhang; Xiaoguang Li; Chaohe Zhang

doi:10.3934/math.2022539

AIMS Mathematics

2022, Volume 7, Issue 6: 9683-9693. doi: 10.3934/math.2022539

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Research article

Instability of standing waves for a quasi-linear Schrödinger equation in the critical case

Xiaoguang Li ^,,
Chaohe Zhang

School of Mathematical Science and V.C. & V.R. Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610068, China

Received: 23 September 2021 Revised: 19 January 2022 Accepted: 19 January 2022 Published: 15 March 2022
MSC : 35F25, 35Q55

We consider the following quasi-linear Schrödinger equation.

$ \begin{align} i\frac{\partial\psi}{\partial t}+\triangle\psi+\psi\triangle|\psi|^2+|\psi|^{p-1}\psi = 0,x\in \mathbb{R}^D, D\geq1, \;\;\;\;\;\;\;\;\;(Q)\end{align} $

where $ \psi: \mathbb{R}^+\times \mathbb{R}^D\rightarrow \mathbb{C} $ is the wave function, $ p = 3+\frac{4}{D} $. It is known that the set of standing waves is stable for $ 1 < p < 3+\frac{4}{D} $ and it is strongly unstable for $ 3+\frac{4}{D} < p < \frac{3D+2}{D-2} $. In this paper, we prove that the standing waves are strongly unstable for $ p = 3+\frac{4}{D} $. Moreover, a property on the set of the ground states of (Q) is investigated.
- quasi-linear Schrödinger equation,
- ground states,
- instability
Citation: Xiaoguang Li, Chaohe Zhang. Instability of standing waves for a quasi-linear Schrödinger equation in the critical case[J]. AIMS Mathematics, 2022, 7(6): 9683-9693. doi: 10.3934/math.2022539

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Abstract

We consider the following quasi-linear Schrödinger equation.

$ \begin{align} i\frac{\partial\psi}{\partial t}+\triangle\psi+\psi\triangle|\psi|^2+|\psi|^{p-1}\psi = 0,x\in \mathbb{R}^D, D\geq1, \;\;\;\;\;\;\;\;\;(Q)\end{align} $

where $ \psi: \mathbb{R}^+\times \mathbb{R}^D\rightarrow \mathbb{C} $ is the wave function, $ p = 3+\frac{4}{D} $. It is known that the set of standing waves is stable for $ 1 < p < 3+\frac{4}{D} $ and it is strongly unstable for $ 3+\frac{4}{D} < p < \frac{3D+2}{D-2} $. In this paper, we prove that the standing waves are strongly unstable for $ p = 3+\frac{4}{D} $. Moreover, a property on the set of the ground states of (Q) is investigated.

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