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

In this paper, we study the Liouville-type equation $${\mathrm{\Delta }}^{2}u-{\lambda }_1\kappa \mathrm{\Delta }u+{\lambda }_2{\kappa }^2(1-{\mathrm{e}}^{4u})=0$$ on a closed Riemannian manifold $(M^4,g)$ with $\mathrm{Ric}⩾3\kappa g$ 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.