Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent
Year: 2026
Type: preprint
Abstract: We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation $Δ^2 u=u^{-7}$ in $\mathbb R^3\setminus\{0\}$, 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 $u(x)\sim(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$ as $x\to 0$; 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 $u(x)\equiv(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$. 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.
Source: arXiv (Cornell University)
Authors: Huang, Xia, Jiang, Yahui, Zhou, Xianmei
Institutions: Zhejiang Normal University, East China Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1609915050, G7270659173
OpenAlex ID: W7213261287