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

We consider the following quasi-linear Schrödinger equation. \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} 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.