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

We consider the focusing energy-critical nonlinear Schrodinger equation Of fourth order iu(t) + Delta(2)u = vertical bar u vertical bar(8/d-4)u, d >= 5, We prove that if a maximal-lifespan radial solution u: I x R d, C obeys sup(t is an element of l) parallel to Delta u(t)parallel to(2) < parallel to Delta W parallel to(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. (C) 2008 Elsevier Inc. All rights reserved.