Uniqueness and symmetry results for solutions of a mean field equation on S{double-struck}2 via a new bubbling phenomenon
Journal
Communications on Pure and Applied Mathematics
Journal Volume
64
Journal Issue
12
Pages
1677-1730
Date Issued
2011
Author(s)
Abstract
Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere S{double-struck}2 involving a Dirac distribution supported at a point P ∈ S{double-struck}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 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.
SDGs
Type
journal article