[Lorsque blow-up ne signifie pas « concentration » : un détour par rapport au résultat de Brézis–Merle]
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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@article{CRMATH_2016__354_5_493_0,
author = {Lin, Chang-Shou and Tarantello, Gabriella},
title = {When {\textquotedblleft}blow-up{\textquotedblright} does not imply {\textquotedblleft}concentration{\textquotedblright}: {A} detour from {Br\'ezis{\textendash}Merle's} result},
journal = {Comptes Rendus. Math\'ematique},
pages = {493--498},
publisher = {Elsevier},
volume = {354},
number = {5},
year = {2016},
doi = {10.1016/j.crma.2016.01.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.01.014/}
}
TY - JOUR AU - Lin, Chang-Shou AU - Tarantello, Gabriella TI - When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result JO - Comptes Rendus. Mathématique PY - 2016 SP - 493 EP - 498 VL - 354 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.01.014/ DO - 10.1016/j.crma.2016.01.014 LA - en ID - CRMATH_2016__354_5_493_0 ER -
%0 Journal Article %A Lin, Chang-Shou %A Tarantello, Gabriella %T When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result %J Comptes Rendus. Mathématique %D 2016 %P 493-498 %V 354 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.01.014/ %R 10.1016/j.crma.2016.01.014 %G en %F CRMATH_2016__354_5_493_0
Lin, Chang-Shou; Tarantello, Gabriella. When “blow-up” does not imply “concentration”: A detour from Brézis–Merle's result. Comptes Rendus. Mathématique, Tome 354 (2016) no. 5, pp. 493-498. doi : 10.1016/j.crma.2016.01.014. http://www.numdam.org/articles/10.1016/j.crma.2016.01.014/
[1] Profile of blow-up solutions to mean field equations with singular data, Commun. Partial Differ. Equ., Volume 29 (2004), pp. 1241-1265
[2] Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Commun. Math. Phys., Volume 229 (2002), pp. 3-47
[3] Uniform estimates and blow-up behavior for solutions of in two dimensions, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 1223-1254
[4] Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Commun. Pure Appl. Math., Volume 55 (2002), pp. 728-771
[5] Topological degree for a mean-field equation on Riemann surfaces, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1667-1727
[6] Mean field equations of Liouville type with singular data: sharper estimates, Discrete Contin. Dyn. Syst., Volume 28 (2010), pp. 1237-1272
[7] Mean field equation of Liouville type with singular data: topological degree, Commun. Pure Appl. Math., Volume 68 (2014), pp. 887-947
[8] Analytic aspects of the Toda system, II: bubbling behavior and existence of solutions, Commun. Pure Appl. Math., Volume 59 (2006), pp. 526-558
[9] Y. Lee, C.S. Lin, J.C. Wei, W. Yang, Degree counting and shadow system for SU(3) Toda system: one bubbling, preprint.
[10] Y. Lee, C.S. Lin, W. Yang, L. Zhang, Degree counting for Toda system with simple singularity: one blowup point, preprint.
[11] Blow up analysis for solutions of in dimension two, Indiana Univ. Math. J., Volume 43 (1994), pp. 1255-1270
[12] Blow-up analysis for Liouville type equations is selfdual gauge fiels vortices, Commun. Contemp. Math., Volume 7 (2005), pp. 177-205
[13] On non-topological solutions of a planar Liouville system of Toda type | arXiv
[14] Analytical, geometrical and topological aspects of mean-field equations on surfaces, Discrete Contin. Dyn. Syst., Ser. A, Volume 28 (2010) no. 3, pp. 931-973
[15] Blow-up analysis for a cosmic string equation | arXiv
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