Vol. 289, No. 1, 2017

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ISSN: 0030-8730
Remarks on GJMS operator of order six

Xuezhang Chen and Fei Hou

Vol. 289 (2017), No. 1, 35–70
DOI: 10.2140/pjm.2017.289.35
Abstract

We study analysis aspects of the sixth-order GJMS operator Pg6. Under conformal normal coordinates around a point, we present the expansions of Green’s function of Pg6 with pole at this point. As a starting point of the study of Pg6, we manage to give some existence results of the prescribed Q-curvature problem on Einstein manifolds. One among them is that for n 10, let (Mn ,g) be a closed Einstein manifold of positive scalar curvature and f a smooth positive function in M. If the Weyl tensor is nonzero at a maximum point of f and f satisfies a vanishing order condition at this maximum point, then there exists a conformal metric g̃ of g such that its Q-curvature Qg̃6 equals f.

Keywords
sixth-order GJMS operator, prescribed $Q$-curvature problem, Green's function, mountain pass critical points
Mathematical Subject Classification 2010
Primary: 53A30, 53C21, 58J05
Secondary: 35B50, 35J08, 35J35
Milestones
Received: 6 March 2016
Revised: 11 October 2016
Accepted: 11 January 2017
Published: 12 May 2017
Authors
Xuezhang Chen
Department of Mathematics & IMS
Nanjing University
210093 Nanjing
China
Fei Hou
Department of Mathematics and IMS
Nanjing University
210093 Nanjing
China