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

DOI10.1016/J.JFA.2022.109649zbMATH Open1507.58016arXiv2012.11785MaRDI QIDQ6356662

Lina Wu, Lei Zhang, Mohameden Ould Ahmedou

Publication date: 21 December 2020

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 $mathbb{R}^4$ 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 emph{isolated simple blow up points}.

Mathematics Subject Classification ID

Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) (58J60) PDEs on manifolds (35R01)

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