On instability of standing waves for the mass-supercritical fractional nonlinear Schr\"odinger equation

We consider the focusing $L^2$-supercritical fractional nonlinear Schr"odinger equation $$i{\mathrm{\partial }}_{t}u-(-\mathrm{\Delta }{)}^{s}u=-\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^d,$$ where $d\geq 2, \frac{d}{2d-1} \leq s <1$and$\frac{4s}{d}<\alpha<\frac{4s}{d-2s}$. By means of the localized virial estimate, we prove that the ground state standing wave is strongly unstable by blow-up. This result is a complement to a recent result of Peng-Shi [J. Math. Phys. 59 (2018), 011508] where the stability and instability of standing waves were studied in the $L^2$-subcritical and $L^2$-critical cases.