Leray-Schauder degree for the resonant Q-curvature problem in even dimensions

TL;DR

This paper develops a precise bubbling rate formula for the resonant Q-curvature problem on even-dimensional manifolds, leading to new existence, compactness, and degree computation results under specific assumptions.

Contribution

It introduces an exact bubbling rate formula using critical points at infinity theory, enabling new existence and degree results for the resonant Q-curvature problem.

Findings

Derived an exact bubbling rate formula for the problem

Established new existence results under positive mass assumption

Computed the Leray-Schauder degree for the equation

Abstract

In this paper, using the theory of critical points at infinity of Bahri, we derive an exact bubbling rate formula for the resonant prescribed Q-curvature equation on closed even-dimensional Riemannian manifolds. Using this, we derive new existence results for the resonant prescribed Q-curvature problem under a positive mass type assumption. Moreover, we derive a compactness theorem for conformal metrics with prescribed Q-curvature under a non-degeneracy assumption. Furthermore, combining the bubbling rate formula with the construction of some blowing-up solutions, we compute the Leray-Schauder degree of the resonant prescribed Q-curvature equation under a non-degeneracy and Morse type assumption.