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Failure of superharmonicity near an isolated singularity of the critical biharmonic equation

  • Juncheng Wei
  • Hui Yang
  • Ronghao Yang
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Abstract

Let n≥5 and u be a positive singular solution of Δ2u=u(n+4)/(n−4) in a punctured ball. A long-standing question is where or not $ -Δu>0$ in the punctured ball. In this paper we show that the answer is negative. More precisely we construct a positive singular solution whose singularity at the origin is nonremovable, while its lower limit there is zero. On a normal subspace, −Δu converges to a strictly negative constant. The spherical minimum has order rβ, where 2<β<4, and the spherical Harnack ratio is unbounded. The proof uses balanced planar polygons, finitely many exterior parameters and a countable gluing construction.

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Cite

BibTeXRIS
BibTeX
@misc{wei2026failure,
  title = {{Failure of superharmonicity near an isolated singularity of the critical biharmonic equation}},
  author = {Juncheng Wei and Hui Yang and Ronghao Yang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05998},
  eprint = {2611.05998},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05998},
}
RIS
TY  - UNPB
TI  - Failure of superharmonicity near an isolated singularity of the critical biharmonic equation
AU  - Juncheng Wei
AU  - Hui Yang
AU  - Ronghao Yang
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05998
UR  - https://arxiv.org/abs/2611.05998
AB  - Let n≥5 and u be a positive singular solution of Δ^2u=u^(n+4)/(n−4) in a punctured ball. A long-standing question is where or not $ -Δu>0$ in the punctured ball. In this paper we show that the answer is negative. More precisely we construct a positive singular solution whose singularity at the origin is nonremovable, while its lower limit there is zero. On a normal subspace, −Δu converges to a strictly negative constant. The spherical minimum has order r^β, where 2<β<4, and the spherical Harnack ratio is unbounded. The proof uses balanced planar polygons, finitely many exterior parameters and a countable gluing construction.
ER  -