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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.) (English. French summary) Zbl 1387.35310

Summary: The pioneering work by H. Brézis and F. Merle [Commun. Partial Differ. Equations 16, No. 8–9, 1223–1253 (1991; Zbl 0746.35006)] 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 H. Ohtsuka and T. Suzuki [Commun. Contemp. Math. 7, No. 2, 177–205 (2005; Zbl 1157.58305)], 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.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B44 Blow-up in context of PDEs
35B45 A priori estimates in context of PDEs
35R01 PDEs on manifolds
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