Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schr"odinger equation of fourth order $i{u}_t+{\mathrm{\Delta }}^{2}u=\mathrm{\mid }u{\mathrm{\mid }}^{\frac{8}{d-4}}u$. We prove that if a maximal-lifespan radial solution $u:I\times {\mathbb{R}}^d\to \mathbb{C}$ obeys ${\displaystyle \underset{t\in I}{\sup }\mathrm{\parallel }\mathrm{\Delta }u(t){\mathrm{\parallel }}_2<\mathrm{\parallel }\mathrm{\Delta }W{\mathrm{\parallel }}_2}$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters.