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

Li Ma; Xingwang Xu

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
[Une estimation uniforme et un résultat de non-existence pour l'équation de Lichnerowicz sur n-espace]

Présenté par : Étienne Ghys

Li Ma ¹ ; Xingwang Xu 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.

Résumé
Abstract

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.

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.

Reçu le : 2009-01-30
Accepté le : 2009-04-09
Publié le : 2009-05-09

DOI : 10.1016/j.crma.2009.04.017

Affiliations des auteurs :

Li Ma ¹ ; Xingwang Xu 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 = {Li Ma and Xingwang Xu},
     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},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.017},
     language = {en},
}

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Li Ma; Xingwang Xu. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.017/

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Cité par

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Cité par Sources :

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

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