Geodesic
Some existence results for the scalar curvature problem via Morse theory
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 4, pp. 267-270.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We prove existence of positive solutions for the equation \( -\triangle_{g_{0}} u + u = (1 + \epsilon K (x)) u^{2^{*}-1} \) 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 \( \epsilon \) 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 \( -\triangle_{g_{0}} u + u = (1 + \epsilon K (x)) u^{2^{*}-1} \) 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 \( \epsilon \) un parametro piccolo. Il problema si riduce a uno studio finito-dimensionale che è affrontato con la teoria di Morse. 
@article{RLIN_1999_9_10_4_a3,
     author = {Malchiodi, Andrea},
     title = {Some existence results for the scalar curvature problem via {Morse} theory},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {267--270},
     publisher = {mathdoc},
     volume = {Ser. 9, 10},
     number = {4},
     year = {1999},
     zbl = {1021.53022},
     mrnumber = {1416171},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_1999_9_10_4_a3/}
}
TY  - JOUR
AU  - Malchiodi, Andrea
TI  - Some existence results for the scalar curvature problem via Morse theory
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 1999
SP  - 267
EP  - 270
VL  - 10
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_1999_9_10_4_a3/
LA  - en
ID  - RLIN_1999_9_10_4_a3
ER  - 
%0 Journal Article
%A Malchiodi, Andrea
%T Some existence results for the scalar curvature problem via Morse theory
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 1999
%P 267-270
%V 10
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_1999_9_10_4_a3/
%G en
%F RLIN_1999_9_10_4_a3
Malchiodi, Andrea. Some existence results for the scalar curvature problem via Morse theory. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 4, pp. 267-270. http://geodesic.mathdoc.fr/item/RLIN_1999_9_10_4_a3/

[1] A. Ambrosetti - M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. Henri Poincaré Analyse Non Linéaire, 15, 1998, 233-252; Preliminary note on C. R. Acad. Sci. Paris, 323, Série I, 1996, 753-758. | fulltext EuDML | fulltext mini-dml | DOI | MR | Zbl

[2] A. Ambrosetti - J. Garcia Azorero - I. Peral, Perturbation of \( - \triangle u + u^{\frac{(N+2)}{(N-2)}} = 0 \), the Scalar Curvature Problem in \( R^{N} \) and related topics. J. Funct. Anal., to appear. | DOI | MR | Zbl

[3] A. Bahri - J. M. Coron, The scalar curvature problem on the standard three dimensional sphere. J. Funct. Anal., 95, 1991, 106-172. | DOI | MR | Zbl

[4] G. Bianchi, The scalar curvature equation on \( \mathbb{R}^{n} \) and on \( S^{n} \). Adv. Diff. Eq., 1, 1996, 857-880. | MR | Zbl

[5] S. A. Chang - P. Yang, Prescribing scalar curvature on \( S^{2} \). Acta Math., 159, 1987, 215-229. | DOI | MR | Zbl

[6] S. A. Chang - P. Yang, Conformal deformation of metrics on \( S^{2} \). J. Diff. Geom., 27, 1988, 259-296. | fulltext mini-dml | MR | Zbl

[7] S. A. Chang - P. Yang, A perturbation result in prescribing scalar curvature on \( S^{2} \). Duke Math. J., 64 (1), 1991, 27-69. | fulltext mini-dml | DOI | MR | Zbl

[8] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, 1993. | MR | Zbl

[9] K. C. Chang - J. Q. Liu, On Nirenberg’s problem. Int. J. Math., 4, 1993, 35-58. | DOI | MR | Zbl

[10] W. X. Chen - W. Ding, Scalar curvature on \( S^{2} \). Trans. Amer. Math. Soc., 303, 1987, 365-382. | Zbl

[11] J. L. Kazdan - F. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. of Math., 101, 1971, 317-331. | MR | Zbl

[12] Y. Y. Li, Prescribing scalar curvature on \( S^{n} \) and related topics, Part 1. J. Diff. Equat., 120, 1995, 319-410. | DOI | MR | Zbl

[13] Y. Y. Li, Prescribing scalar curvature on \( S^{n} \) and related topics, Part 2, Existence and compactness. Comm. Pure Appl. Math., 49, 1996, 437-477. | DOI | MR | Zbl

[14] A. Malchiodi, The Scalar Curvature Problem on \( S^{n} \): an approach via Morse Theory. Preprint SISSA. | DOI | MR | Zbl

[15] R. Schoen - D. Zhang, Prescribed scalar curvature on the \( n \)-sphere. Calc. Var., 4, 1996, 1-25. | DOI | MR | Zbl