Some existence results for the scalar curvature problem via Morse theory

Malchiodi, Andrea

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 10 (1999), p. 267-270 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

We prove existence of positive solutions for the equation $- △_{g_{0}} u + u = (1 + ϵ K (x)) u^{2^{*} - 1}$ on $S^{n}$ , arising in the prescribed scalar curvature problem. is the Laplace-Beltrami operator on $S^{n}$ , $2^{*}$ is the critical Sobolev exponent, and $ϵ$ is a small parameter. The problem can be reduced to a finite dimensional study which is performed with Morse theory.

Si dimostra l’esistenza di soluzioni positive per l’equazione $- △_{g_{0}} u + u = (1 + ϵ K (x)) u^{2^{*} - 1}$ su $S^{n}$ , che nasce del problema della curvatura scalare prescritta. è l’operatore di Laplace-Beltrami su $S^{n}$ , $2^{*}$ è l’esponente critico di Sobolev, ed $ϵ$ un parametro piccolo. Il problema si riduce a uno studio finito-dimensionale che è affrontato con la teoria di Morse.

Publié le : 1999-12-01

MR 1767933 | Zbl 1021.53022

@article{RLIN_1999_9_10_4_267_0,
     author = {Andrea Malchiodi},
     title = {Some existence results for the scalar curvature problem via Morse theory},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {10},
     year = {1999},
     pages = {267-270},
     zbl = {1021.53022},
     mrnumber = {1767933},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1999_9_10_4_267_0}
}

Malchiodi, Andrea. Some existence results for the scalar curvature problem via Morse theory. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 10 (1999) pp. 267-270. http://gdmltest.u-ga.fr/item/RLIN_1999_9_10_4_267_0/

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