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Given $(M,g)$, a smooth compact Riemannian manifold of dimension $n \ge 5$, we investigate compactness for the fourth order geometric equation $P_gu = u^{2^\sharp-1}$, where $P_g$ is the Paneitz operator, and $2^\sharp = 2n/(n-4)$ is critical from the Sobolev viewpoint. We prove that the equation is compact when the Paneitz operator is of strong positive type.

Copyright 2004 American Mathematical Society
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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 135-141
• Publisher Identifier: S 1079-6762(04)00138-6
• 2000 Mathematics Subject Classification. Primary: 58E30, 58J05
• Key words and phrases. Blow-up behavior, compactness, Paneitz operator
• Received by editors October 7, 2004
• Posted on December 10, 2004
• Communicated by Tobias Colding
• Comments (When Available)

Emmanuel Hebey

Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

E-mail address: Emmanuel.Hebey@math.u-cergy.fr

Frédéric Robert

Laboratoire J.A.Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France

E-mail address: frobert@math.unice.fr