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
;
2016-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
