Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d⩾5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)‖2<‖ΔW‖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.