On stability and instability of standing waves for the nonlinear Schr\"odinger equation with inverse-square potential

We consider the focusing nonlinear Schr"odinger equation with inverse square potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+c\mathrm{\mid }x{\mathrm{\mid }}^{-2}u=-\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,u(0)=u_0\in H^1,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^d,$$ where $d\ge 3$, $c\mathrm{\ne }0$, $c<\lambda (d)={\left(\frac{d-2}{2}\right)}^2$ 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.