Liouville theorem of the subcritical biharmonic equation on complete manifolds

TL;DR

This paper proves a Liouville theorem for positive solutions of a subcritical biharmonic equation on complete manifolds with nonnegative Ricci curvature, showing nonexistence under certain conditions.

Contribution

It introduces a differential identity using invariant tensors and establishes a second-order derivative estimate for the biharmonic equation on manifolds.

Findings

No positive solutions exist for the specified equation when n≥5 and 1<α<(n+4)/(n-4).

Derived a differential identity using invariant tensors.

Established a second-order derivative estimate via Bernstein's technique.

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.