The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds

Abstract

In this paper, we study the Liouville-type equation \[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). A key step in our proof is a second-order derivative estimate, which is established via the continuity method. As an application of the classification results, we derive an Onofri-type inequality on the 4-sphere and prove its rigidity.