Some existence results for the scalar curvature problem via Morse theory

Andrea Malchiodi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 4, page 267-270
  • ISSN: 1120-6330

Abstract

top
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.

How to cite

top

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 10.4 (1999): 267-270. <http://eudml.org/doc/252447>.

@article{Malchiodi1999,
abstract = {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.},
author = {Malchiodi, Andrea},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elliptic equations; Critical exponent; Scalar curvature; Perturbation method; Morse theory; elliptic equations; critical Sobolev exponent; scalar curvature; perturbation method},
language = {eng},
month = {12},
number = {4},
pages = {267-270},
publisher = {Accademia Nazionale dei Lincei},
title = {Some existence results for the scalar curvature problem via Morse theory},
url = {http://eudml.org/doc/252447},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Malchiodi, Andrea
TI - Some existence results for the scalar curvature problem via Morse theory
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/12//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 4
SP - 267
EP - 270
AB - 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.
LA - eng
KW - Elliptic equations; Critical exponent; Scalar curvature; Perturbation method; Morse theory; elliptic equations; critical Sobolev exponent; scalar curvature; perturbation method
UR - http://eudml.org/doc/252447
ER -

References

top
  1. Ambrosetti, A. - Badiale, M., 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. Zbl1004.37043MR1416171DOI10.1016/S0294-1449(97)89300-6
  2. Ambrosetti, A. - Garcia Azorero, J. - Peral, I., Perturbation of - △ ⁢ u + u N + 2 N - 2 = 0 , the Scalar Curvature Problem in R N and related topics. J. Funct. Anal., to appear. Zbl0938.35056MR1696454DOI10.1006/jfan.1999.3390
  3. Bahri, A. - Coron, J. M., The scalar curvature problem on the standard three dimensional sphere. J. Funct. Anal., 95, 1991, 106-172. Zbl0722.53032MR1087949DOI10.1016/0022-1236(91)90026-2
  4. Bianchi, G., The scalar curvature equation on R n and on S n . Adv. Diff. Eq., 1, 1996, 857-880. Zbl0865.35044MR1392008
  5. Chang, S. A. - Yang, P., Prescribing scalar curvature on S 2 . Acta Math., 159, 1987, 215-229. Zbl0636.53053MR908146DOI10.1007/BF02392560
  6. Chang, S. A. - Yang, P., Conformal deformation of metrics on S 2 . J. Diff. Geom., 27, 1988, 259-296. Zbl0649.53022MR925123
  7. Chang, S. A. - Yang, P., A perturbation result in prescribing scalar curvature on S 2 . Duke Math. J., 64 (1), 1991, 27-69. Zbl0739.53027MR1131392DOI10.1215/S0012-7094-91-06402-1
  8. Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, 1993. Zbl0779.58005MR1196690
  9. Chang, K. C. - Liu, J. Q., On Nirenberg’s problem. Int. J. Math., 4, 1993, 35-58. Zbl0786.58010MR1209959DOI10.1142/S0129167X93000042
  10. Chen, W. X. - Ding, W., Scalar curvature on S 2 . Trans. Amer. Math. Soc., 303, 1987, 365-382. Zbl0635.35026
  11. Kazdan, J. L. - Warner, F., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. of Math., 101, 1971, 317-331. Zbl0297.53020MR375153
  12. Li, Y. Y., Prescribing scalar curvature on S n and related topics, Part 1. J. Diff. Equat., 120, 1995, 319-410. Zbl0827.53039MR1347349DOI10.1006/jdeq.1995.1115
  13. Li, Y. Y., Prescribing scalar curvature on S n and related topics, Part 2, Existence and compactness. Comm. Pure Appl. Math., 49, 1996, 437-477. Zbl0849.53031MR1383201DOI10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
  14. Malchiodi, A., The Scalar Curvature Problem on S n : an approach via Morse Theory. Preprint SISSA. Zbl1012.53035MR1911824DOI10.1007/s005260100110
  15. Schoen, R. - Zhang, D., Prescribed scalar curvature on the n -sphere. Calc. Var., 4, 1996, 1-25. Zbl0843.53037MR1379191DOI10.1007/BF01322307

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.