Uniform bound and a non-existence result for Lichnerowicz equation in the whole <i>n</i>-space

Ma, Li; Xu, Xingwang

doi:10.1016/j.crma.2009.04.017

Differential Geometry/Mathematical Physics

Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space

Presented by: Étienne Ghys

Ma, Li ¹; Xu, Xingwang ²

¹ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
² Mathematics Department, The National University of Singapore, 10, Kent Ridge Crescent, Singapore 119260

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 805-808.

Abstract
Résumé

In this Note, we give a uniform bound and a non-existence result for positive solutions to the Lichnerowicz equation in $R^{n}$ . In particular, we show that positive smooth solutions to:

Δ u + f (u) = 0, u > 0, in R^{n}

where

f (u) = u^{- p - 1} - u^{p - 1},

are uniformly bounded.

Dans cette Note, nous donnons une estimation uniforme et un résultat de non-existence pour les solutions positives de l'équation de Lichnerowicz sur $R^{n}$ . En particulier, nous montrons que les solutions positives régulières de :

Δ u + f (u) = 0, u > 0, dans R^{n}

où

f (u) = u^{- p - 1} - u^{p - 1},

sont bornées.

Received: 2009-01-30
Accepted: 2009-04-09
Published online: 2009-05-09

DOI: 10.1016/j.crma.2009.04.017

Author's affiliations:

Ma, Li ¹; Xu, Xingwang ²

¹ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
² Mathematics Department, The National University of Singapore, 10, Kent Ridge Crescent, Singapore 119260

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     author = {Ma, Li and Xu, Xingwang},
     title = {Uniform bound and a non-existence result for {Lichnerowicz} equation in the whole \protect\emph{n}-space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {805--808},
     publisher = {Elsevier},
     volume = {347},
     number = {13-14},
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     doi = {10.1016/j.crma.2009.04.017},
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Ma, Li; Xu, Xingwang. Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 805-808. doi : 10.1016/j.crma.2009.04.017. http://www.numdam.org/articles/10.1016/j.crma.2009.04.017/

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Cited by Sources:

^☆ The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.