Classification and asymptotics for the boundary Yamabe equation with an isolated singularity
Abstract
We classify positive singular solutions of the boundary Yamabe equation on the punctured half-space in every dimension n≥3. Previous symmetry results reduce this problem to a nonlinear PDE in two variables, for which classical phase-plane methods are unavailable. We develop a Hamiltonian method that combines local bifurcation with compactness to prove that every nonstationary cylindrical profile of a singular solution is periodic and uniquely determined, up to translation, by its Hamiltonian energy. At each stationary or periodic profile, the space of solutions of the linearized equation is two-dimensional. Finally, every positive solution in a half-ball with a nonremovable isolated boundary singularity is asymptotic to a unique global singular solution, with an algebraically decaying relative error.
Related
- Failure of superharmonicity near an isolated singularity of the critical biharmonic equation
- A Parametrization of Solutions to the Yamabe Equation with Isolated Singularity
- Classification and long-time asymptotics of (2,2)-sign solitons for the damped nonlinear Klein-Gordon equation
- The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II
- Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities
BibTeX
@misc{jin2026classification,
title = {{Classification and asymptotics for the boundary Yamabe equation with an isolated singularity}},
author = {Tianling Jin and Yuxuan Liao and Jingang Xiong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07636},
eprint = {2611.07636},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07636},
}
RIS
TY - UNPB
TI - Classification and asymptotics for the boundary Yamabe equation with an isolated singularity
AU - Tianling Jin
AU - Yuxuan Liao
AU - Jingang Xiong
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07636
UR - https://arxiv.org/abs/2611.07636
AB - We classify positive singular solutions of the boundary Yamabe equation on the punctured half-space in every dimension n≥3. Previous symmetry results reduce this problem to a nonlinear PDE in two variables, for which classical phase-plane methods are unavailable. We develop a Hamiltonian method that combines local bifurcation with compactness to prove that every nonstationary cylindrical profile of a singular solution is periodic and uniquely determined, up to translation, by its Hamiltonian energy. At each stationary or periodic profile, the space of solutions of the linearized equation is two-dimensional. Finally, every positive solution in a half-ball with a nonremovable isolated boundary singularity is asymptotic to a unique global singular solution, with an algebraically decaying relative error.
ER -