Blow-up analysis and compactness for critical Dirac equations on πn
Abstract
We study blow-up and compactness for the critical nonlinear Dirac equation Dg0Οi = Hi|Οi|2/(nβ1)Οi on πn, nβ₯ 3, where g0 is the round metric and Hiβ H>0 in C0(πn). Under a first-threshold bound on the critical mass, we prove that every blowing-up sequence has, after passing to a subsequence, a unique blow-up point p satisfying H(p)=maxπnH. The rescaled solutions converge to a ground-state Dirac bubble that carries the entire limiting critical mass. Under local uniform C1 bounds on Hi, we use the Kelvin transform and a second rescaling to prove that p is an isolated simple blow-up point. We also derive sharp pointwise and gradient estimates in cylindrical coordinates and identify the normalized Green limit. If the coefficients converge in C2 near p, the Pohozaev identity yields Hessg0H(p)=0. Consequently, if a positive Hβ C2(πn) satisfies Ξg0H(p)<0 at every global maximum, then solutions with critical mass bounded by a level slightly above the first threshold are uniformly bounded in Lβ(πn). This bound remains uniform under small C2 perturbations of H.
Related
- Kinetic SDEs and kinetic Fokker-Planck equations with critical Morrey-type drifts, and local blow-ups
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- The CR Nirenberg Problem, Part I: Blow-up Analysis and Compactness
BibTeX
@misc{li2026blowup,
title = {{Blow-up analysis and compactness for critical Dirac equations on $\mathbb S^n$}},
author = {Ting Li and Zhongwei Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33507},
eprint = {2610.33507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33507},
}
RIS
TY - UNPB
TI - Blow-up analysis and compactness for critical Dirac equations on π^n
AU - Ting Li
AU - Zhongwei Tang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33507
UR - https://arxiv.org/abs/2610.33507
AB - We study blow-up and compactness for the critical nonlinear Dirac equation D_g_0Ο_i = H_i|Ο_i|^2/(nβ1)Ο_i on π^n, nβ₯ 3, where g_0 is the round metric and H_iβ H>0 in C^0(π^n). Under a first-threshold bound on the critical mass, we prove that every blowing-up sequence has, after passing to a subsequence, a unique blow-up point p satisfying H(p)=max_π^nH. The rescaled solutions converge to a ground-state Dirac bubble that carries the entire limiting critical mass. Under local uniform C^1 bounds on H_i, we use the Kelvin transform and a second rescaling to prove that p is an isolated simple blow-up point. We also derive sharp pointwise and gradient estimates in cylindrical coordinates and identify the normalized Green limit. If the coefficients converge in C^2 near p, the Pohozaev identity yields Hess_g_0H(p)=0. Consequently, if a positive Hβ C^2(π^n) satisfies Ξ_g_0H(p)<0 at every global maximum, then solutions with critical mass bounded by a level slightly above the first threshold are uniformly bounded in L^β(π^n). This bound remains uniform under small C^2 perturbations of H.
ER -