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

Abstract

We consider the focusing nonlinear Schrödinger equation with inverse square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^αu, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d\ge 3$, $c\ne 0$, $c<\lambda (d)={\left(\frac{d-2}{2}\right)}^2$ and $0<\alpha \le \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<\alpha <\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $\alpha =\frac{4}{d}$, we show that ground state standing waves are strongly unstable by blow-up.

20 pages, submitted