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

Abstract

We consider the focusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|\Delta u(t)\|_{2}<\|\Delta W\|_{2}$, 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.