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

In this Note, we present a complete classification of singularities of positive solutions of the equation Δu+μ|x|2u=h(u) in Ω{0}, where Ω is a bounded domain of RN, N3, 0Ω, and 0<μ<(N2)24. The case μ=0 with h(t)=tq, q>1 were treated by Brezis and Véron.

Dans cette Note, nous présentons une classification complète des singularités de solutions positives de l'équation Δu+μ|x|2u=h(u) dans Ω{0}, où Ω est un domaine borné de RN, N3, 0Ω, et où 0<μ<(N2)24. Le cas μ=0 avec h(t)=tq, q>1 a été traité par Brezis et Véron.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2008.12.018

Nirmalendu Chaudhuri 1 ; Florica C. Cîrstea 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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Nirmalendu Chaudhuri; Florica C. Cîrstea. On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 153-158. doi : 10.1016/j.crma.2008.12.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.018/

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

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

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

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

[7] F.C. Cîrstea; V. Rădulescu 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] D. Gilbarg; N. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983

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

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

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

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

[13] L. Véron 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

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Cité par 17 documents. Sources : zbMATH

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

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