Asymptotic profile for the sub-extremals of the sharp Sobolev inequality on the sphere

Communications in Partial Differential Equations

Volumn 26, Issue 5-6, 2001, Pages 743-778

  Druet, Olivier ^a   Robert, Frédéric ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0012080216     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-100002377     Document Type: Article

References (29)

1. F.V. Atkinson and L.A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70, 1987, 349-365.
2. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geometry, 11, 1976, 573-598.
3. T. Aubin, Nonlinear analysis on manifolds, Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften, vol. 252, Berlin, Springer-Verlag, 1982.
4. H. Berestycki, L. Nirenberg and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. on Pure and Applied Math., 47, 1994, 47-92.
5. M.F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Inv. Math., 106, 1991, 489-539.
6. H. Brézis and L.A. Peletier, Asymptotics for elliptic equations involving critical Sobolev exponents, In PDE and the calculus of variations; F. Colombini, A. Marino, L. Modica and S. Spagnolo Editors, Basel, Birkhäuser, 1989.
7. L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with Sobolev growth. Comm. on Pure Applied Math., 42, 1989, 271-297.
8. Z. Djadli and O. Druet, Extremal functions for optimal Sobolev inequalities on compact manifolds, Calc. Var., 2000, to appear.
9. O. Druet, The best constants problem in Sobolev inequalities, Math. Ann., 314, 1999, 327-346.
10. B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. on Pure and Applied Math., 34, 1981, 525-598.
11. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, 1977.
12. Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H Poincaré, Analyse non-linéaire, 8, 1991, 159-174.
13. E. Hebey, Changements de métriques conformes sur la sphère. Le problème de Nirenberg, Bull. Sc. Math., 114, 1990, 215-242.
14. E. Hebey, La méthode d'isométries-concentration dans le cas d'un problème non-linéaire sur les variétés compactes à bord avec exposant critique de Sobolev, Bull. Sc. Math., 116, 1992, 35-51.
15. E. Hebey, Asymptotics for some quasilinear elliptic equations, Diff. Int. Eq., 9, 1996, 71-88.
16. E. Hebey, Fonctions extrémales pour une inégalité de Sobolev optimale dans la classe conforme de la sphère, J. Math. Pures et App. 77, 1998, 721-733.
17. E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, Vol. 5, Courant Institute of Mathematical Sciences, 1999.
18. E. Hebey, Asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth, Diff. Int. Eq., to appear.
19. E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. Journal, 79, 1995, 235-279.
20. E. Hebey and M. Vaugon, Meilleures constantes dans le théorème d'inclusion de Sobolev, Ann. Inst. H. Poincaré, Analyse non-linéaire, vol. 13, 1996, 57-93.
21. J.L. Kazdan and F.W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Diff. Geometry, 10, 1975, 113-134.
22. n and related problems, Part I, J. Diff. Eq., 120, 1995, 319-410.
23. M. Obata, The conjectures on conformal transformations on Riemannian manifolds, J. Diff. Geometry, 110, 1975, 247-258.
24. n involving critical Sobolev exponents, J. Diff. Eq., 99, 1992, 78-107.
25. O. Rey, Proof of two conjectures of H. Brézis and L.A. Peletier, Manus. Math., 65, 1989, 19-37.
26. F. Robert, Etude asymptotique d'une équation non-linéaire à croissance de Sobolev critique. Le cas radial, Prépublications de l'Université de Cergy-Pontoise, 19, 1999.
27. R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture Notes in Mathematics, 1365, Springer-Verlag, Berlin, 1989.
28. R. Schoen and Zhang, D. Prescribed scalar curvature on the n-sphere, Calc. Var. 4, 1996, 1-25.
29. G. Talenti, Best constants in Sobolev inequality, Ann. di Matem. Pura ed Appl., 110, 1976, 353-372.