On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

Abstract: We consider the focusing $L^2$-supercritical fractional nonlinear Schr\"odinger equation [ i\partial_t u - (-\Delta)^s u = -|u|^\alpha u, \quad (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.