Uniqueness and symmetry results for solutions of a mean field equation on 𝕊2 via a new bubbling phenomenon |
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Authors: | Daniele Bartolucci Chang‐Shou Lin Gabriella Tarantello |
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Affiliation: | 1. University of Rome “Tor Vergata”, Department of Mathematics, Via della ricerca scientifica n.1, 00133 Rome, ITALY;2. National Taiwan University, Taida Institute for Mathematical Sciences, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106 |
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Abstract: | Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere 𝕊2 involving a Dirac distribution supported at a point P ∈ 𝕊2. 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 ) 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. |
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