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

Abstract

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order i u t + Δ 2 u = | u | 8 d − 4 u , d ⩾ 5 . We prove that if a maximal-lifespan radial solution u : I × R d → C obeys sup t ∈ I ‖ Δ u ( t ) ‖ 2 < ‖ Δ 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.