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

We consider the focusing L 2-supercritical fractional nonlinear Schrödinger equation i∂tu − (−∆) s u = −|u| α u, (t, x) ∈ R + × R d , where d ≥ 2, d 2d−1 ≤ s < 1 and 4s d < α < 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