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 two-sphere involving a Dirac distribution supported at a point P of S^2. As needed for the applications, we show that solutions ”concentrate” exactly at 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. Indeed, we need to rule out the possibility that, after blow up (in a suitable scale), the solution sequence may admit a double ”peak” profile described in terms of appropriate “limiting” problems. For instance we need to account for the presence of non-axially symmetric solutions in such ”limiting” problems. In this process, we establish a symmetry result about a maximal circle through P and its antipodal point P*, that applies to more general situations where the full axial symmetry cannot be expected.