Uniqueness and symmetry results for solutions of a mean field equation on 𝕊 2 via a new bubbling phenomenon

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.