On instability of radial standing waves for the nonlinear Schrödinger equation with inverse-square potential

Abstract

We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^α u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\ge 3$, $u:\mathbb{R}\times {\mathbb{R}}^d\to \mathbb{C}$, $c\ne 0$ satisfies $c<\lambda (d):={\left(\frac{d-2}{2}\right)}^2$ and $\frac{4}{d}<\alpha <\frac{4}{d-2}$. This result extends a recent result of Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential}, \url{arXiv:1805.01245}] where the stability and instability of standing waves were shown in the $L^2$-subcritical and $L^2$-critical cases.

15 pages, submitted

References in corpus (3)

• Suppression of the quantum collapse in binary bosonic gases
• Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential
• Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schrödinger equation with inverse-square potential