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PreprintCC BY-NC-SA 4.0arXiv

The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

  • Hua-Yang Wang
  • Jingang Xiong
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Abstract

Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay type log-periodic solutions posed by del Pino--Musso--Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron--Ponce--Véron (2007). We construct a global continuum of positive log-periodic solutions containing the local bifurcation branch and prove that blow-up along this continuum occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scaling-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli--Gidas--Spruck (1989).

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Cite

BibTeXRIS
BibTeX
@misc{wang2026critical,
  title = {{The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II}},
  author = {Hua-Yang Wang and Jingang Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.16106},
  eprint = {2608.16106},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.16106},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II
AU  - Hua-Yang Wang
AU  - Jingang Xiong
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2608.16106
UR  - https://arxiv.org/abs/2608.16106
AB  - Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay type log-periodic solutions posed by del Pino--Musso--Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron--Ponce--Véron (2007). We construct a global continuum of positive log-periodic solutions containing the local bifurcation branch and prove that blow-up along this continuum occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scaling-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli--Gidas--Spruck (1989).
ER  -