Comptes Rendus
Partial Differential Equations
Bubbling solutions for an anisotropic Emden–Fowler equation
Comptes Rendus. Mathématique, Volume 343 (2006) no. 4, pp. 253-258.

We consider the anisotropic Emden–Fowler equation: (a(x)u)+ε2a(x)eu=0 in Ω, u=0 on ∂Ω where ΩR2 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

can approach to +∞ as ε0. These results show a striking difference with the isotropic case (a(x)constant).

On considère l'équation de Emden–Fowler anisotropique : (a(x)u)+ε2a(x)eu=0 dans Ω, u=0 sur ∂ΩΩR2 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

peut approcher +∞ quand ε0. Ces résultats montrent une différence frappante avec le cas isotropique (a(x)constante).

Published online:
DOI: 10.1016/j.crma.2006.05.017

Juncheng Wei 1; Dong Ye 2; Feng Zhou 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
     author = {Juncheng Wei and Dong Ye and Feng Zhou},
     title = {Bubbling solutions for an anisotropic {Emden{\textendash}Fowler} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {253--258},
     publisher = {Elsevier},
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     year = {2006},
     doi = {10.1016/j.crma.2006.05.017},
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Juncheng Wei; Dong Ye; Feng Zhou. Bubbling solutions for an anisotropic Emden–Fowler equation. Comptes Rendus. Mathématique, Volume 343 (2006) no. 4, pp. 253-258. doi : 10.1016/j.crma.2006.05.017.

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