Non-compactness of the prescribed Q-curvature problem in large dimensions

Juncheng Wei; Chunyi Zhao

doi:10.1007/s00526-011-0477-9

Non-compactness of the prescribed Q-curvature problem in large dimensions

Juncheng Wei^*
, Chunyi Zhao

^*Corresponding author for this work

School of Mathematical Sciences

Chinese University of Hong Kong

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Q_g be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to P_g(u) = cu^N+4/N-4, u>0 on M where P_g is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.

Original language	English
Pages (from-to)	123-164
Number of pages	42
Journal	Calculus of Variations and Partial Differential Equations
Volume	46
Issue number	1-2
DOIs	https://doi.org/10.1007/s00526-011-0477-9
State	Published - Jan 2013

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abstract = "Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Qg be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to Pg(u) = cuN+4/N-4, u>0 on M where Pg is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.",

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N2 - Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Qg be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to Pg(u) = cuN+4/N-4, u>0 on M where Pg is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.

AB - Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Qg be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to Pg(u) = cuN+4/N-4, u>0 on M where Pg is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.

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JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

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ER -

Non-compactness of the prescribed Q-curvature problem in large dimensions

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