On the existence of blowing-up solutions for a mean field equation

Angela Pistoia; Pierpaolo Esposito; Massimo Grossi

doi:10.1016/j.anihpc.2004.12.001

JournalsaihpcVol. 22, No. 2pp. 227–257

On the existence of blowing-up solutions for a mean field equation

Angela Pistoia
Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via Scarpa, 16, 00166, Roma, Italy
Pierpaolo Esposito
Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy
Massimo Grossi
Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro, 00166, Roma, Italy

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Abstract

In this paper we construct single and multiple blowing-up solutions to the mean field equation:;

{- Δ u = λ \frac{V ( x ) e ^{u}}{\int _{Ω} V ( x ) e ^{u}} u = 0 in Ω, on \partial Ω,

where $Ω$ is a smooth bounded domain in $R^{2}$ , $V$ is a smooth function positive somewhere in $Ω$ and $λ$ is a positive parameter.

Résumé

Dans ce papier nous construisons des solutions qui explosent pour l'équation de champ moyen :

{- Δ u = λ \frac{V ( x ) e ^{u}}{\int _{Ω} V ( x ) e ^{u}} u = 0 in Ω, on \partial Ω,

où $Ω$ est un domaine borné dans $R^{2}$ , $V$ est une fonction positive dans $Ω$ et $λ$ est un paramètre positif.

Cite this article

Angela Pistoia, Pierpaolo Esposito, Massimo Grossi, On the existence of blowing-up solutions for a mean field equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 2, pp. 227–257

DOI 10.1016/J.ANIHPC.2004.12.001