Liouville theorem of the subcritical biharmonic equation on complete manifolds

2025/08/20 by Xi‐Nan Ma, Tian Wu, Ma, Xi-Nan +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2508.14497

openalex publication_date 2025/08/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the subcritical biharmonic equation Δ² u=u^α on a complete, connected, and non-compact Riemannian manifold (Mⁿ,g) with nonnegative Ricci curvature. Using the method of invariant tensors, we derive a differential identity to obtain a Liouville theorem, i.e., there is no positive C⁴ solution if n\geqslant5 and 1<α<(n+4)/(n-4). We establish a crucial second-order derivative estimate, which is established via Bernstein's technique and the continuity method.

Citations

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Cited by

• On a new region for the Lane-Emden conjecture in higher dimensions

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