The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher

TL;DR

This paper proves that solutions to the focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher are global and scatter if their energy is below that of the ground state, extending previous results to higher dimensions.

Contribution

It extends the scattering and global existence results for the focusing energy-critical NLS to dimensions five and higher without symmetry assumptions.

Findings

Solutions with energy below the ground state are global and scatter.

Bounded kinetic energy blow-up solutions must concentrate at least the ground state energy.

Results generalize previous work in lower dimensions and symmetric cases.

Abstract

We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabla W\|_2$, 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. 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 \cite{Evian, kenig-merle} for spherically symmetric initial data and dimensions $d=3,4,5$.