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Uniqueness and symmetry results for solutions of a mean field equation on 𝕊 2 via a new bubbling phenomenon
Author(s) -
Bartolucci Daniele,
Lin ChangShou,
Tarantello Gabriella
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20385
Subject(s) - uniqueness , singularity , mathematics , dirac measure , dirac (video compression format) , field (mathematics) , symmetry (geometry) , mathematical analysis , mathematical physics , dirac equation , physics , pure mathematics , quantum mechanics , geometry , dirac algebra , neutrino , dirac spinor
Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere 2 involving a Dirac distribution supported at a point P ∈ 2 . Consistently with the physical applications, we show that solutions “concentrate” precisely around the point P for some limiting value of a given parameter. We use this fact to obtain symmetry (about the axis $\overrightarrow {OP}$ ) and uniqueness property for the solution. The presence of the Dirac measure makes such a task particularly delicate to handle from the analytical point of view. In fact, the bubbling phenomenon about the singularity allows the existence of solution sequences with a double‐peak profile near P . The new and more delicate part of this paper is to exclude this possibility by using the method of moving planes together with the Alexandrov‐Bol inequality. © 2011 Wiley Periodicals, Inc.