Sharp Estimates for Bubbling Solutions to Some Fourth-Order Geometric Equations

In this article, we derive an analogue of the exact bubbling rate formula of Chen-Lin [19] for the Q-curvature equation on closed four-dimensional Riemannian manifolds. Using this, we derive new existence results for the prescribed Q-curvature problem on closed four-dimensional Riemannian manifolds under a positive mass type assumption. Furthermore, as an other application, we deduce a compactness theorem for conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds under a nondegeneracy assumption. Our method is of variational nature and is based on the tools of critical points theory at infinity of Bahri [3]. Our exact bubbling rate formula is also used in a subsequent paper to interpret the index counting formula in [54] as the Leray-Schauder degree of the corresponding equation.