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.

Tarantello, G., & Lin, C. (2016). When "Blow-up" does not imply "concentration" : a detour from Brezis-Merle's result. COMPTES RENDUS MATHÉMATIQUE, 354, 493-498 [10.1016/j.crma.2016.01.014].

When "Blow-up" does not imply "concentration" : a detour from Brezis-Merle's result.

Tarantello G
;
2016

Abstract

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.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
English
Tarantello, G., & Lin, C. (2016). When "Blow-up" does not imply "concentration" : a detour from Brezis-Merle's result. COMPTES RENDUS MATHÉMATIQUE, 354, 493-498 [10.1016/j.crma.2016.01.014].
Tarantello, G; Lin, C
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/198716
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