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

Abstract

<abstract><p>We consider the following quasi-linear Schrödinger equation.</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document} $ \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} $ \end{document} </tex-math> </disp-formula></p> <p>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 &lt; p &lt; 3+\frac{4}{D} $ and it is strongly unstable for $ 3+\frac{4}{D} &lt; p &lt; \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.</p></abstract>