article published yes Rowan Killip author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca 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. Johns Hopkins University Press2010 eng 0002-9327 1080-6377 0804.101810.1353/ajm.0.0107 1322361-424 yes Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>, vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a>. R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,” <i>American Journal of Mathematics</i>, vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010. Killip, R., &#38; Vişan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>. Johns Hopkins University Press. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a> R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424. Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” <i>American Journal of Mathematics</i>. Johns Hopkins University Press, 2010. <a href="https://doi.org/10.1353/ajm.0.0107">https://doi.org/10.1353/ajm.0.0107</a>. Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 132(2), 361–424. Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. <i>American Journal of Mathematics</i>. 2010;132(2):361-424. doi:<a href="https://doi.org/10.1353/ajm.0.0107">10.1353/ajm.0.0107</a> 220232026-06-19T07:31:39Z2026-06-19T10:24:15Z