Positive solutions for slightly super-critical elliptic equations in contractible domains

Molle, Riccardo; Passaseo, Donato

doi:10.1016/S1631-073X(02)02502-5

Positive solutions for slightly super-critical elliptic equations in contractible domains
[Solutions positives pour l'équation Δu+u^{(n+2)/(n−2)+ε}=0 en ouverts contractibles]

Molle, Riccardo ¹ ; Passaseo, Donato ²

¹ Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
² Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy

Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 459-462.

Résumé
Abstract

On donne des exemples d'ouverts bornés $Ω$ , même contractibles, satisfaisant la propriété suivante : il existe $\bar{k} (Ω)$ tel que, pour tout $k ⩾ \bar{k} (Ω)$ , le problème $P (ϵ, Ω)$ ci-dessous, pour ε>0 suffisamment petit, a des solutions qui pour ε→0 explosent exactement en k points. On prouve aussi que ces points convergent vers des points de $\partial Ω$ quand k→∞.

We give examples of bounded domains $Ω$ , even contractible, having the following property: there exists $\bar{k} (Ω)$ such that, for every integer $k ⩾ \bar{k} (Ω)$ , problem $P (ϵ, Ω)$ below, for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. We also prove that the blow-up points tend to some points of $\partial Ω$ as k→∞.

Reçu le : 2002-07-08
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02502-5

Affiliations des auteurs :

Molle, Riccardo ¹ ; Passaseo, Donato ²

@article{CRMATH_2002__335_5_459_0,
     author = {Molle, Riccardo and Passaseo, Donato},
     title = {Positive solutions for slightly super-critical elliptic equations in contractible domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--462},
     publisher = {Elsevier},
     volume = {335},
     number = {5},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02502-5},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02502-5/}
}

TY  - JOUR
AU  - Molle, Riccardo
AU  - Passaseo, Donato
TI  - Positive solutions for slightly super-critical elliptic equations in contractible domains
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 459
EP  - 462
VL  - 335
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02502-5/
DO  - 10.1016/S1631-073X(02)02502-5
LA  - en
ID  - CRMATH_2002__335_5_459_0
ER  -

%0 Journal Article
%A Molle, Riccardo
%A Passaseo, Donato
%T Positive solutions for slightly super-critical elliptic equations in contractible domains
%J Comptes Rendus. Mathématique
%D 2002
%P 459-462
%V 335
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02502-5/
%R 10.1016/S1631-073X(02)02502-5
%G en
%F CRMATH_2002__335_5_459_0

Molle, Riccardo; Passaseo, Donato. Positive solutions for slightly super-critical elliptic equations in contractible domains. Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 459-462. doi : 10.1016/S1631-073X(02)02502-5. http://www.numdam.org/articles/10.1016/S1631-073X(02)02502-5/

Bibliographie
Cité par

[1] Bahri, A.; Coron, J.M. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., Volume 41 (1988), pp. 253-294

[2] Bahri, A.; Li, Y.; Rey, O. On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var., Volume 3 (1995) no. 1, pp. 67-93

[3] Brezis, H.; Peletier, L.A. Asymptotics for elliptic equations involving critical growth (Colombini; Modica; Spagnolo, eds.), P.D.E. and the Calculus of Variations, Birkhäuser, Basel, 1989, pp. 149-192

[4] Coron, J.M. Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris, Série I, Volume 299 (1984) no. 7, pp. 209-212

[5] Dancer, E.N. A note on an equation with critical exponent, Bull. London Math. Soc., Volume 20 (1988) no. 6, pp. 600-602

[6] M. Del Pino, P. Felmer, M. Musso, Multipeak solutions for super-critical elliptic problems in domains with small holes, Preprint

[7] Ding, W.Y. Positive solutions of Δu+u^{(n+2)/(n−2)}=0 on contractible domains, J. Partial Differential Equations, Volume 2 (1989) no. 4, pp. 83-88

[8] Han, Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 8 (1991) no. 2, pp. 159-174

[9] Kazdan, J.; Warner, F.W. Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., Volume 28 (1975) no. 5, pp. 567-597

[10] R. Molle, D. Passaseo, (to appear)

[11] Pohožaev, S.I. On the eigenfunctions of the equation Δu+λf(u)=0, Soviet Math. Dokl., Volume 6 (1965), pp. 1408-1411

[12] Passaseo, D. Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math., Volume 65 (1989) no. 2, pp. 147-165

[13] Passaseo, D. Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., Volume 114 (1993) no. 1, pp. 97-105

[14] Passaseo, D. Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke Math. J., Volume 92 (1998) no. 2, pp. 429-457

[15] Rey, O. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Volume 89 (1990) no. 1, pp. 1-52

Cité par Sources :