Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent
Abstract
We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation Δ2 u=u−7 in ℝ3∖{0}, which extend continuously to the origin with u(0)=0. First, we prove that if the associated conformal metric g=u−4|dx|2 has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile u(x)∼(4/3)1/4|x|1/2 as x→ 0; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition u(x)=o(|x|) at infinity forces u(x)≡(4/3)1/4|x|1/2. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.
Related
- Existence of solutions for a nonlinear elliptic equation involving (p(x),q(x))-Laplacian and (p(x),q(x))-biharmonic Operator
- Existence Results for the Critical Biharmonic Elliptic Equation Involving Singular Nonlinearities
- Failure of superharmonicity near an isolated singularity of the critical biharmonic equation
- A Liouville type rigidity result for entire solutions of the k-Hessian equation
- Rigidity for Convex Entire Solutions to the k-Hessian Equation
BibTeX
@misc{huang2026rigidity,
title = {{Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent}},
author = {Xia Huang and Yahui Jiang and Xianmei Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13820},
eprint = {2609.13820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13820},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent
AU - Xia Huang
AU - Yahui Jiang
AU - Xianmei Zhou
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13820
UR - https://arxiv.org/abs/2609.13820
AB - We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation Δ^2 u=u^−7 in ℝ^3∖{0}, which extend continuously to the origin with u(0)=0. First, we prove that if the associated conformal metric g=u^−4|dx|^2 has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile u(x)∼(4/3)^1/4|x|^1/2 as x→ 0; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition u(x)=o(|x|) at infinity forces u(x)≡(4/3)^1/4|x|^1/2. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.
ER -