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## Compactness and global estimates
for the geometric

Paneitz equation in high dimensions

### Emmanuel Hebey and Frédéric Robert

**Abstract.**
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
*

The copyright for this article reverts to public domain after 28 years from publication.

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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`

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