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

Nirmalendu Chaudhuri; Florica C. Cîrstea

doi:10.1016/j.crma.2008.12.018

Partial Differential Equations

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

Presented by: Haïm Brezis

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

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

Abstract (Original language)
Résumé

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.

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.

Received: 2008-10-16
Accepted: 2008-12-17
Published online: 2009-02-01

DOI: 10.1016/j.crma.2008.12.018

Author's affiliations:

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

@article{CRMATH_2009__347_3-4_153_0,
     author = {Nirmalendu Chaudhuri and Florica C. C{\^\i}rstea},
     title = {On trichotomy of positive singular solutions associated with the {Hardy{\textendash}Sobolev} operator},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {153--158},
     publisher = {Elsevier},
     volume = {347},
     number = {3-4},
     year = {2009},
     doi = {10.1016/j.crma.2008.12.018},
     language = {en},
}

TY  - JOUR
AU  - Nirmalendu Chaudhuri
AU  - Florica C. Cîrstea
TI  - On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 153
EP  - 158
VL  - 347
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2008.12.018
LA  - en
ID  - CRMATH_2009__347_3-4_153_0
ER  -

%0 Journal Article
%A Nirmalendu Chaudhuri
%A Florica C. Cîrstea
%T On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator
%J Comptes Rendus. Mathématique
%D 2009
%P 153-158
%V 347
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2008.12.018
%G en
%F CRMATH_2009__347_3-4_153_0

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/

References
Cited by

[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

Cited by Sources:

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

Comments - Policy