Classification and a priori estimates for the singular prescribing Q-curvature equation on 4-manifold

Abstract

On (M,g) a compact riemannian 4-manifold we consider the prescribed Q-curvature equation defined on M with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on R4 and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are isolated simple blow up points.