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 u(k) satisfying [GRAPHICS] where (i) alpha(k) is an element of R+, alpha(k)-->alpha; (ii) V-k, is an element of C-0,C-1(B-1), 0 < a less than or equal to V-k less than or equal to b and \DeltaV(k)\ less than or equal to A in B-1. We prove that necessarily, beta is an element of 8piN boolean OR {8pi(1 + alpha) + 8piZ(+)}. The result is "sharp" as shown by explicit examples, and should be compared with that obtained by Li-Shafrir (Ind. Univ. Math. J. 43(4) (1994) 1255) concerning the situation where alpha(k) = 0, For Allk is an element of N. (C) 2004 Elsevier Inc. All rights reserved.