Bubbling solutions for an anisotropic Emden–Fowler equation

Wei, Juncheng; Ye, Dong; Zhou, Feng

doi:10.1016/j.crma.2006.05.017

Partial Differential Equations

Bubbling solutions for an anisotropic Emden–Fowler equation
[Solutions à bulles pour une équation de Emden–Fowler anisotropique]

Présenté par : Haïm Brezis

Wei, Juncheng ¹ ; Ye, Dong ² ; Zhou, Feng ³

¹ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
² Département de Mathématiques, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France
³ Department of Mathematics, East China Normal University, Shanghai 200062, China

Comptes Rendus. Mathématique, Tome 343 (2006) no. 4, pp. 253-258.

Résumé
Abstract

On considère l'équation de Emden–Fowler anisotropique : $\nabla (a (x) \nabla u) + ε^{2} a (x) e^{u} = 0$ dans Ω, $u = 0$ sur ∂Ω où $Ω \subset R^{2}$ est un domaine régulier borné et a est une fonction régulière strictement positive. Nous étudions l'effet du coefficient anisotropique $a (x)$ sur l'existence des solutions à bulles. Nous montrons que pour un maximum local strict de la fonction a, il existe des solutions avec un nombre arbitraire de bulles. Par conséquent, la quantité

T_{ε} = ε^{2} \int_{Ω} a (x) e^{u} d x

peut approcher +∞ quand

ε \to 0

. Ces résultats montrent une différence frappante avec le cas isotropique (

a (x) \equiv constante

We consider the anisotropic Emden–Fowler equation: $\nabla (a (x) \nabla u) + ε^{2} a (x) e^{u} = 0$ in Ω, $u = 0$ on ∂Ω where $Ω \subset R^{2}$ is a smooth bounded domain and $a (x)$ is a positive, smooth function. We investigate the effect of anisotropic coefficient $a (x)$ on the existence of bubbling solutions. We show that at given strict local maximum points of a, there exist solutions with arbitrarily many bubbles. As a consequence, the quantity

T_{ε} = ε^{2} \int_{Ω} a (x) e^{u} d x

can approach to +∞ as

ε \to 0

. These results show a striking difference with the isotropic case (

a (x) \equiv constant

Reçu le : 2006-05-11
Publié le : 2006-07-20

DOI : 10.1016/j.crma.2006.05.017

Affiliations des auteurs :

Wei, Juncheng ¹ ; Ye, Dong ² ; Zhou, Feng ³

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     title = {Bubbling solutions for an anisotropic {Emden{\textendash}Fowler} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {253--258},
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Wei, Juncheng; Ye, Dong; Zhou, Feng. Bubbling solutions for an anisotropic Emden–Fowler equation. Comptes Rendus. Mathématique, Tome 343 (2006) no. 4, pp. 253-258. doi : 10.1016/j.crma.2006.05.017. http://www.numdam.org/articles/10.1016/j.crma.2006.05.017/

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