On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator

Chaudhuri, Nirmalendu; Cîrstea, Florica C.

doi:10.1016/j.crma.2008.12.018

Partial Differential Equations

On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator
[Sur la trichotomie des solutions positives singulières associées à l'opérateur de Hardy–Sobolev]

Présenté par : Haïm Brezis

Chaudhuri, Nirmalendu ¹ ; Cîrstea, Florica C.²

¹ School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
² School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Comptes Rendus. Mathématique, Tome 347 (2009) no. 3-4, pp. 153-158.

Résumé
Abstract

Dans cette Note, nous présentons une classification complète des singularités de solutions positives de l'équation $Δ u + \frac{μ}{{| x |}^{2}} u = h (u)$ dans $Ω ∖ {0}$ , où Ω est un domaine borné de $R^{N}$ , $N ⩾ 3$ , $0 \in Ω$ , et où $0 < μ < \frac{{(N - 2)}^{2}}{4}$ . Le cas $μ = 0$ avec $h (t) = t^{q}$ , $q > 1$ a été traité par Brezis et Véron.

In this Note, we present a complete classification of singularities of positive solutions of the equation $Δ u + \frac{μ}{{| x |}^{2}} u = h (u)$ in $Ω ∖ {0}$ , where Ω is a bounded domain of $R^{N}$ , $N ⩾ 3$ , $0 \in Ω$ , and $0 < μ < \frac{{(N - 2)}^{2}}{4}$ . The case $μ = 0$ with $h (t) = t^{q}$ , $q > 1$ were treated by Brezis and Véron.

Reçu le : 2008-10-16
Accepté le : 2008-12-17
Publié le : 2009-02-01

DOI : 10.1016/j.crma.2008.12.018

Affiliations des auteurs :

Chaudhuri, Nirmalendu ¹ ; Cîrstea, Florica C. ²

¹ School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
² School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

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     title = {On trichotomy of positive singular solutions associated with the {Hardy{\textendash}Sobolev} operator},
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Chaudhuri, Nirmalendu; Cîrstea, Florica C. On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator. Comptes Rendus. Mathématique, Tome 347 (2009) no. 3-4, pp. 153-158. doi : 10.1016/j.crma.2008.12.018. http://www.numdam.org/articles/10.1016/j.crma.2008.12.018/

Bibliographie
Cité par

[1] Adimurthi, O.; Chaudhuri, N.; Ramaswamy, M. An improved Hardy–Sobolev inequality and its application, Proc. Amer. Math. Soc., Volume 130 (2002), pp. 489-505

[2] Brezis, H.; Oswald, L. Singular solutions for some semilinear elliptic equations, Arch. Rational Mech. Anal., Volume 99 (1987), pp. 249-259

[3] Brezis, H.; Vázquez, J.L. Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, Volume 10 (1997), pp. 443-469

[4] Brezis, H.; Véron, L. Removable singularities of some nonlinear elliptic equations, Arch. Rational Mech. Anal., Volume 75 (1980), pp. 1-6

[5] N. Chaudhuri, F.C. Cîrstea, On classification of isolated singularities of solutions associated with the Hardy–Sobolev operator, in preparation

[6] Cîrstea, F.C.; Du, Y. Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity, J. Funct. Anal., Volume 250 (2007), pp. 317-346

[7] Cîrstea, F.C.; Rădulescu, V. Extremal singular solutions for degenerate logistic-type equations in anisotropic media, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 119-124

[8] Gilbarg, D.; Trudinger, N. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983

[9] Guerch, B.; Véron, L. Local properties of stationary solutions of some nonlinear singular Schrödinger equations, Rev. Mat. Iberoamericana, Volume 7 (1991), pp. 65-114

[10] Pucci, P.; Serrin, J. The strong maximum principle revisited, J. Differential Equations, Volume 196 (2004), pp. 1-66

[11] Seneta, E. Regularly Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, Berlin, Heidelberg, 1976

[12] Véron, L. Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. T.M.A., Volume 5 (1981), pp. 225-242

[13] Véron, L. Weak and strong singularities of nonlinear elliptic equations, Berkeley, CA, 1983 (Proc. Sympos. Pure Math.), Volume vol. 45, Amer. Math. Soc., Providence, RI (1986), pp. 477-795

Cité par Sources :

^☆ This work were partially supported by an Australian Research Council Grant of Professors Neil Trudinger and Xu-Jia Wang.