A quantization property for blow up solutions of singular Liouville-type equations

Motivated by the study of self-dual vortices in gauge field theory (cf. (Vortices and Monopoles, Birkhauser, Boston, 1980; Solitons in Field Theory and Nonlinear Analysis, Monographs in Mathematics, Springer, New York, 2001)), we consider a “concentrating”solution-sequence uk satisfying-Δuk=|z|2αkVk(z)eukinB1,|z|2αkVk(z)euk⇀βδz=0weaklyinthesenseofmeasuresinB1,where