Bubbling solutions for an anisotropic Emden–Fowler equation

Juncheng Wei; Dong Ye; Feng Zhou

doi:10.1016/j.crma.2006.05.017

Partial Differential Equations

Bubbling solutions for an anisotropic Emden–Fowler equation

Presented by: Haïm Brezis

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

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

Abstract
Résumé

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

).

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

).

Received: 2006-05-11
Published online: 2006-07-20

DOI: 10.1016/j.crma.2006.05.017

Author's affiliations:

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

@article{CRMATH_2006__343_4_253_0,
     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},
     volume = {343},
     number = {4},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.017},
     language = {en},
}

TY  - JOUR
AU  - Juncheng Wei
AU  - Dong Ye
AU  - Feng Zhou
TI  - Bubbling solutions for an anisotropic Emden–Fowler equation
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 253
EP  - 258
VL  - 343
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2006.05.017
LA  - en
ID  - CRMATH_2006__343_4_253_0
ER  -

%0 Journal Article
%A Juncheng Wei
%A Dong Ye
%A Feng Zhou
%T Bubbling solutions for an anisotropic Emden–Fowler equation
%J Comptes Rendus. Mathématique
%D 2006
%P 253-258
%V 343
%N 4
%I Elsevier
%R 10.1016/j.crma.2006.05.017
%G en
%F CRMATH_2006__343_4_253_0

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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.017/

References
Cited by

[1] H. Brezis; F. Merle Uniform estimates and blow-up behavior for solutions of $- Δ u = V (x) e^{u}$ in two dimensions, Comm. Partial Differential Equations, Volume 16 (1991) no. 8–9, pp. 1223-1253

[2] S. Baraket; F. Pacard Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, Volume 6 (1998) no. 1, pp. 1-38

[3] S. Chanilo; Y.Y. Li Continuity of solutions of uniformly elliptic equations in $R^{2}$ , Manuscripta Math., Volume 77 (1992), pp. 415-433

[4] X. Chen Remarks on the existence of branch bubbles on the blowup-analysis of equation $- Δ u = e^{2 u}$ in dimension two, Comm. Anal. Geom., Volume 7 (1999), pp. 295-302

[5] M. Del Pino; M. Kowalczyk; M. Musso Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, Volume 24 (2005), pp. 47-81

[6] M. Del Pino; P. Felmer Semiclassical states for nonlinear Schrödinger equations, J. Funct. Anal., Volume 149 (1997), pp. 245-265

[7] P. Esposito; M. Grossi; A. Pistoia On the existence of blowing-up solutions for a mean field equation, Ann. I. H. Poincaré Anal. Nonlinéaire, Volume 22 (2005), pp. 227-257

[8] L. Ma; J. Wei Convergence for a Liouville equation, Comm. Math. Helv., Volume 76 (2001), pp. 506-514

[9] N. Mizoguchi; T. Suzuki Equations of gas combustion: S-shaped bifurcation and mushrooms, J. Differential Equations, Volume 134 (1997), pp. 183-215

[10] K. Nagasaki; Y. Suzuki Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., Volume 3 (1990), pp. 173-188

[11] J. Wei, D. Ye, F. Zhou, Bubbling solutions for an anisotropic Emden–Fowler equation, Calc. Var. Partial Differential Equations (2006), in press

[12] D. Ye Une remarque sur le comportement asymptotique des solutions de $- Δ u = λ f (u)$ , C. R. Acad. Sci. Paris I, Volume 325 (1997), pp. 1279-1282

[13] D. Ye; F. Zhou A generalized two dimensional Emden–Fowler equation with exponential nonlinearity, Calc. Var. Partial Differential Equations, Volume 13 (2001), pp. 141-158

Cited by Sources:

Comments - Policy