Existence of bubbling solutions without mass concentration

Lee, Youngae; Lin, Chang-Shou; Yang, Wen

doi:10.5802/aif.3261

Existence of bubbling solutions without mass concentration
[Existence de solutions bouillonnantes sans concentration de masse]

Lee, Youngae ¹ ; Lin, Chang-Shou ² ; Yang, Wen ³

¹ Department of Mathematics Education Teachers College Kyungpook National University Daegu (South Korea)
² Taida Institute for Mathematical Sciences Center for Advanced Study in Theoretical Sciences National Taiwan University Taipei 106 (Taiwan)
³ Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences P.O. Box 71010 Wuhan 430071 (P. R. China)

Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 895-940.

Résumé
Abstract

Le travail séminal de Brezis et Merle a été pionnier dans l’étude des phénomènes de bulles de l’équation du champ moyen avec des sources singulières. Lorsque les points de vortex ne s’affaissent pas, l’équation du champ moyen possède la propriété de ce que l’on appelle « le bouillonnement implique une concentration de masse » . Récemment, Lin et Tarantello ont remarqué que les phénomènes de « bouillonnement implique une concentration de masse » pourraient ne pas s’appliquer en général s’il y a effondrement des singularités se produit. Dans cet article, nous construisons le premier exemple concret de solution bulleuse non concentrée de l’équation du champ moyen avec des singularités d’effondrement.

The seminal work by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the so-called “bubbling implies mass concentration”. Recently, Lin and Tarantello pointed out that the “bubbling implies mass concentration” phenomena might not hold in general if the collapse of singularities occurs. In this paper, we shall construct the first concrete example of non-concentrated bubbling solution of the mean field equation with collapsing singularities.

Reçu le : 2017-08-11
Accepté le : 2017-03-27
Publié le : 2019-06-03

Zbl

DOI : 10.5802/aif.3261

Classification : 53A30, 35B44, 35J15, 82D55
Keywords: bubbling phenomena, mean field equation
Mot clés : phénomènes de bulles, équation de champ moyen

Affiliations des auteurs :

Lee, Youngae ¹ ; Lin, Chang-Shou ² ; Yang, Wen ³

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Lee, Youngae; Lin, Chang-Shou; Yang, Wen. Existence of bubbling solutions without mass concentration. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 895-940. doi : 10.5802/aif.3261. http://www.numdam.org/articles/10.5802/aif.3261/

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