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

We consider the focusing L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-supercritical fractional nonlinear Schrödinger equation i∂tu-(-Δ)su=-|u|αu,(t,x)∈R+×Rd,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} i\partial _t u - (-\varDelta )^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb {R}^+ \times \mathbb {R}^d, \end{aligned}$$\end{document}where d≥2,d2d-1≤s<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2, \frac{d}{2d-1} \le s <1$$\end{document} and 4sd<α<4sd-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{4s}{d}<\alpha <\frac{4s}{d-2s}$$\end{document}. By means of the localized virial estimate, we prove that the ground-state standing wave is strongly unstable by blowup. This result is a complement to a recent result of Peng–Shi (J Math Phys 59:011508, 2018) where the stability and instability of standing waves were studied in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-subcritical and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-critical cases.