Uniqueness and symmetry results for solutions of a mean field equation on 𝕊² via a new bubbling phenomenon

Abstract

Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere S 2 involving a Dirac distribution supported at a point P 2 S 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.

The sphere covering inequality and its applications

“…arise in many important problems in mathematics, mathematical physics and biology. Such equations have been extensively studied in the context of Moser-Trudinger inequalities, Chern-Simons self-dual vortices, Toda systems, conformal geometry, statistical mechanics of two-dimensional turbulence, self-gravitating cosmic strings, theory of elliptic functions and hyperelliptic curves and free boundary models of cell motility, see [BFR,BL,BL2,BLT,BCLT,BT,BT2,Be,CLS,CaY,CLMP,CLMP2,CK,CFL,CY,CY2,CY3,DJLW,GL,L,L2,LM,LM2,LW,LW2,LWY,Y] and the references cited therein. The sphere covering inequality was recently introduced in [GM], and has been applied to solve various problems about symmetry and uniqueness of solutions of elliptic equations with exponential nonlinearity in dimension n = 2.…”

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

“…The latter equation (and its counterpart on manifolds, see (15) below) has been widely discussed in the last decades since it arises in several problems of mathematics and physics, such as Electroweak and Chern-Simons self-dual vortices [65,67,75], conformal geometry on surfaces [71,46,24,25], statistical mechanics of two-dimensional turbulence [20] and of self-gravitating systems [74] and cosmic strings [61], theory of hyperelliptic curves [22], Painlevé equations [27] and Moser-Trudinger inequalities [18,37,40,45,60]. There are by now many results concerning existence and multiplicity [3,7,8,9,11,15,16,21,30,32,34,35,36,54,57,58], uniqueness [12,13,14,42,44,50,52,66], blow-up phenomena [6,10,17,19,…”

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

“…and let η * be the rearrangement of η as given in (14). Finally, let F (s) and P (s) be defined as in the proof of Proposition 2.1 which we recall here for reader's convenience,…”