On the Brezis–Nirenberg problem on $ {\mathbf{S}}^{3}$, and a conjecture of Bandle–Benguria

Chen, Wenyi; Wei, Juncheng

doi:10.1016/j.crma.2005.06.001

Partial Differential Equations

On the Brezis–Nirenberg problem on

S^{3}

, and a conjecture of Bandle–Benguria
[Sur l'équation de Brezis–Nirenberg sur

S^{3}

et une conjecture de Bandle–Benguria]

Présenté par : Haïm Brezis

Chen, Wenyi ¹ ; Wei, Juncheng ²

¹ Department of Mathematics, Wuhan University, Wuhan, Hubei 430072, PR China
² Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Comptes Rendus. Mathématique, Tome 341 (2005) no. 3, pp. 153-156.

Résumé
Abstract

Nous considérons le problème de Brezis–Nirenberg suivant sur $S^{3}$

- Δ_{S^{3}} u = λ u + u^{5} dans D, u > 0 dans D et u = 0 sur \partial D,

où D est une boule géodésique sur

S^{3}

de rayon géodésique

θ_{1}

, et

- Δ_{S^{3}}

est l'opérateur de Laplace–Beltrami sur

S^{3}

. Nous montrons que pour tout

λ < - \frac{3}{4}

et tout

θ_{1} < π

avec

π - θ_{1}

suffisamment petit (dependant de λ), il existe des solutions pour le problème précédent. Ce résultat répond à une conjecture de Bandle et Benguria [J. Differential Equations 178 (2002) 264–279] et de Brezis et Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394].

We consider the following Brezis–Nirenberg problem on $S^{3}$

- Δ_{S^{3}} u = λ u + u^{5} in D, u > 0 in D and u = 0 on \partial D,

where D is a geodesic ball on

S^{3}

with geodesic radius

θ_{1}

, and

Δ_{S^{3}}

is the Laplace–Beltrami operator on

S^{3}

. We prove that for any

λ < - \frac{3}{4}

and for every

θ_{1} < π

with

π - θ_{1}

sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394].

Reçu le : 2005-05-31
Publié le : 2005-08-01

DOI : 10.1016/j.crma.2005.06.001

Affiliations des auteurs :

Chen, Wenyi ¹ ; Wei, Juncheng ²

¹ Department of Mathematics, Wuhan University, Wuhan, Hubei 430072, PR China
² Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

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     title = {On the {Brezis{\textendash}Nirenberg} problem on $ {\mathbf{S}}^{3}$, and a conjecture of {Bandle{\textendash}Benguria}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {153--156},
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Chen, Wenyi; Wei, Juncheng. On the Brezis–Nirenberg problem on $ {\mathbf{S}}^{3}$, and a conjecture of Bandle–Benguria. Comptes Rendus. Mathématique, Tome 341 (2005) no. 3, pp. 153-156. doi : 10.1016/j.crma.2005.06.001. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.001/

Bibliographie
Cité par

[1] Bandle, C.; Benguria, R. The Brezis–Nirenberg problem on $S^{3}$ , J. Differential Equations, Volume 178 (2002), pp. 264-279

[2] Bandle, C.; Peletier, L.A. Emden–Fowler equations for the critical exponent in $S^{3}$ , Math. Ann., Volume 313 (1999), pp. 83-93

[3] Brezis, H.; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983), pp. 437-477

[4] Brezis, H.; Peletier, L.A. Elliptic equations with critical exponent on $S^{3}$ : new non-minimizing solutions, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 291-394

[5] del Pino, M.; Felmer, P.; Musso, M. Two-bubble solutions in the super-critical Bahri–Coron's problem, Calc. Var. Partial Differential Equations, Volume 16 (2003), pp. 113-145

[6] Gui, C.; Wei, J. Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differential Equations, Volume 158 (1999), pp. 1-27

[7] Struwe, M. Variational Methods and Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990

[8] Rey, O.; Wei, J. Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity, Part I: $N = 3$ , J. Funct. Anal., Volume 212 (2004), pp. 472-499

[9] S.I. Stingelin, Das Brezis–Nirenberg auf der Sphäre $S^{N}$ , Inauguraldissertation, Univerität Basel, 2004

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