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arXiv · 2508.14497

Liouville theorem of the subcritical biharmonic equation on complete manifolds

Xi-Nan Ma·Tian Wu·Wangzhe Wu
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Abstract

In this paper, we study the subcritical biharmonic equation \[\Delta ^2 u=u^\alpha\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ 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 $C^4$ solution if $n\geqslant5$ and $1<\alpha<\frac{n+4}{n-4}$. We establish a crucial second-order derivative estimate, which is established via Bernstein's technique and the continuity method.

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Xi-Nan Ma, Tian Wu, Wangzhe Wu. 2025-08-20. Liouville theorem of the subcritical biharmonic equation on complete manifolds. https://arxiv.org/abs/2508.14497

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