Liouville theorem of the subcritical biharmonic equation on complete manifolds

In this paper, we study the subcritical biharmonic equation $${\mathrm{\Delta }}^{2}u=u^{\alpha }$$ 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 $n⩾5$ 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.