Comptes Rendus
no. 5
Partial differential equations
When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result
[Lorsque blow-up ne signifie pas « concentration » : un détour par rapport au résultat de Brézis–Merle]

Présenté par : Haïm Brézis

Chang-Shou Lin1 ; Gabriella Tarantello2
1 Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Rome, Italy
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 493-498.

Le travail pionnier de Brézis–Merle [3] appliqué aux équations de champ moyen de type Liouville (1) (voir ci-dessous) implique que toute suite non bornée (blow-up suite) montre un nombre fini de points (points de blow-up) autour desquels leur masse se concentre. Dans cette note, nous donnons quelques exemples de blow-up suites qui ne satisfont pas cette conclusion, dans le sens où leur masse s'étale au moment où on considère des situations qui s'écartent légèrement des hypothèses du travail de Brézis–Merle. La presence de masse « residuelle » dans les phénomènes de blow-up avait été remarquée auparavant par Ohtsuka–Suzuki [12] ; en revanche, aucun example explicite n'avait été proposé. Par rapport au system de Toda, ce nouveau phénomène apparaît plutôt naturellement et rend le calcul du degré de Leray–Schauder plus difficile que la résolution de la simple équation de champ moyen.

The pioneering work by Brézis–Merle [3] applied to mean-field equations of Liouville type (1) (see below) implies that any unbounded sequence of solutions (i.e. a sequence of blow-up solutions) must exhibit only finitely many points (blow-up points) around which their “mass” concentrate. In this note, we describe some examples of blow-up solutions that violate such conclusion, in the sense that their mass may spread, as soon as we consider situations which mildly depart from Brézis–Merle's assumptions. The presence of a “residual” mass in blow-up phenomena was pointed out by Ohtsuka–Suzuki in [12], although such possibility was not substantiated by any explicit examples. We mention that for systems of Toda-type, this new phenomenon occurs rather naturally and it makes the calculation of the Leray Schauder degree much harder than the resolution of the single mean-field equation.

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Accepté le :
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DOI : 10.1016/j.crma.2016.01.014
Chang-Shou Lin 1 ; Gabriella Tarantello 2

1 Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Rome, Italy 
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     journal = {Comptes Rendus. Math\'ematique},
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Chang-Shou Lin; Gabriella Tarantello. When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 493-498. doi : 10.1016/j.crma.2016.01.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.014/

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