PreprintarXiv licencearXiv
Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds
Abstract
We study the critical biharmonic equation Δ2 u=u(n+4)/(n−4), on a complete, connected, and non-compact Riemannian manifold (Mn,g) of dimension n≥ 5 with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space ℝn, and the solution is given by u(x)= ( [λ2 n(n−4)(n2−4)](n−4)/8 )/((1+λ|x−x0|2)(n−4)/2), λ>0, x0∈ℝn .
Related
- Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the L2-critical level
- On the classification of solutions to the Logarithmic Laplacian critical Choquard equation
- Nonexistence of solutions to Δpu+Δqu+us|∇ u|t≤ 0 on geodesically complete noncompact Riemannian manifolds
- Continuous solutions to the quaternionic Monge-Ampère equation on locally flat HKT manifolds
- Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds
BibTeX
@misc{li2026classification,
title = {{Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds}},
author = {Huabin Li and Tian Wu and Xiao Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07657},
eprint = {2609.07657},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07657},
}
RIS
TY - UNPB
TI - Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds
AU - Huabin Li
AU - Tian Wu
AU - Xiao Zhou
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07657
UR - https://arxiv.org/abs/2609.07657
AB - We study the critical biharmonic equation Δ^2 u=u^(n+4)/(n−4), on a complete, connected, and non-compact Riemannian manifold (M^n,g) of dimension n≥ 5 with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space ℝ^n, and the solution is given by u(x)= ( [λ^2 n(n−4)(n^2−4)]^(n−4)/8 )/((1+λ|x−x_0|^2)^(n−4)/2), λ>0, x_0∈ℝ^n .
ER -