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

Abstract: In this paper, we study the Liouville-type equation [\Delta ^2 u-\lambda_1\kappa\Delta u+\lambda_2\kappa^2(1-\mathrm e^{4u})=0] on a closed Riemannian manifold ((M^4,g)) with (\operatorname{Ric}\geqslant 3\kappa g) and (\kappa>0). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters (\lambda_1,\lambda_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.