On the Classification of N-Points Concentrating Solutions for Mean Field Equations and the Symmetry Properties of the N-Vortex Singular Hamiltonian on the Unit Disk

Motivated by the analysis of the multiple bubbling phenomenon (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7-8):1241-1265, 2004) for a singular mean field equation on the unit disk (Bartolucci and Montefusco in Nonlinearity 19:611-631, 2006), for any Na parts per thousand yen3 we characterize a subset of the 2 pi/N-symmetric part of the critical set of the N-vortex singular Hamiltonian. In particular we prove that this critical subset is of saddle type. As a consequence of our result, and motivated by a recently posed open problem (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7-8):1241-1265, 2004), we can prove the existence of a multiple bubbling sequence of solutions for the singular mean field equation.