Finite-mass soliton-type rigidity and four-channel reduction for the three-dimensional nonradial focusing energy-critical nonlinear Schrödinger equation
arXiv:2607.25735
The paper classifies possible minimal blow‑up scenarios for the 3‑D nonradial focusing energy‑critical nonlinear Schrödinger equation and proves that any finite‑mass, bounded‑scale almost‑periodic solution must be identically zero, leaving only a residual quasi‑soliton channel as a potential counterexample to scattering.
Abstract
The concentration--compactness channels below the ground-state threshold are investigated for the three-dimensional nonradial focusing energy-critical nonlinear Schrödinger equation. After one-sided normalization, a minimal critical element falls into four classes: the finite-time, rapid-cascade, bounded-scale finite-mass, and residual quasi-soliton channels. The first three classes are rigorously excluded. The main result shows, without radial symmetry, zero momentum, or a slow spatial center, that every finite-mass bounded-scale almost-periodic solution is identically zero. Consequently, any minimal counterexample to below-threshold scattering must lie in the residual quasi-soliton channel; if its scale is bounded, then it has infinite mass at every time.