Normalized ground states for the fractional nonlinear Schr\"{o}dinger equations

In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr\"{o}dinger equation \begin{equation} i\partial_{t}\psi=(-\Delta)^{s}\psi-f(\psi), \qquad (0.1)\end{equation} where $0<s<1$, $f(\psi)=|\psi|^{p}\psi$ with $\frac{4s}{N}<p<\frac{4s}{N-2s}$ or $f(\psi)=(|x|^{-\gamma}\ast|\psi|^2)\psi$ with $2s<\gamma<\min\{N,4s\}$. To this end, we look for normalized solutions of the associated stationary equation \begin{equation} (-\Delta)^s u+\omega u-f(u)=0. \qquad (0.2) \end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.