@article{22023,
  abstract     = {We consider the focusing energy-critical nonlinear Schrödinger equation iut + ∆u = −|u|
4 d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan solution u : I × Rd → C obeys supt∈I k∇u(t)k2 < k∇Wk2, 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. We also show that any solution that blows up with bounded kinetic
energy must concentrate at least the kinetic energy of the ground state. Similar
results were obtained by Kenig and Merle in [17, 18] for spherically symmetric
initial data and dimensions d = 3, 4, 5.},
  author       = {Killip, Rowan and Visan, Monica},
  issn         = {1080-6377},
  journal      = {American Journal of Mathematics},
  number       = {2},
  pages        = {361--424},
  publisher    = {Johns Hopkins University Press},
  title        = {{The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher}},
  doi          = {10.1353/ajm.0.0107},
  volume       = {132},
  year         = {2010},
}