We study the strong instability of ground state standing waves for the focusing $L^2$-supercritical NLS with inverse-square potential $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+c|x{|}^{-2}u=-|u{|}^{\alpha }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$ where $d\ge 3$, $u:\mathbb{R}\times {\mathbb{R}}^d\to \mathbb{C}$, $c\ne 0$ satisfies $c<\lambda \left(d\right):={\left(\left(d-2\right)/2\right)}^2$ and $4/d<\alpha <4/\left(d-2\right)$. This result extends a recent result of Bensouilah et al. [On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential. J. Math. Phys. 59 (2018), 101505] where the stability and instability of standing waves were shown in the $L^2$-subcritical and $L^2$-critical cases.

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具有反平方势的非线性薛定谔方程的驻波不稳定性

我们研究了用于聚焦的基态驻波的强不稳定性 ${升}^2$-具有平方反比势的超临界 NLS $\mathrm{一世}{\mathrm{\partial }}_{吨}你+\mathrm{\Delta }你+C|X{|}^{-2}你=-|你{|}^{\alpha }你,(吨,X)\in \mathbb{电阻}\times {\mathbb{电阻}}^d,$ 在哪里 $d\ge 3$, $你：\mathbb{电阻}\times {\mathbb{电阻}}^d\to \mathbb{C}$, $C\ne 0$ 满足 $C<\lambda \left(d\right):={\left(\left(d-2\right)/2\right)}^2$ 和 $4/d<\alpha <4/\left(d-2\right)$. 该结果扩展了 Bensouilah 等人的最新结果。[关于具有反平方势的非线性薛定谔方程驻波的稳定性和不稳定性。J. 数学。物理。59 (2018), 101505] 其中驻波的稳定性和不稳定性显示在${升}^2$-亚临界和 ${升}^2$- 危急情况。