The Liouville equation with singular data: A concentration-compactness principle via a local representation formula

For a bounded domain Omega subset of R-2, we establish a concentration-compactness result for the following class of "singular" Lionville equations: -Deltau = e(u) - 4piSigma(j=1)(m)alpha(j)delta(pj) in Omega, where p(j) is an element of Omega, alpha(j) > 0 and delta(pj) denotes the Dirac measure with pole at point p(j), j = 1,...,m. Our result extends Brezis-Merle's theorem (Comm. Partial Differential Equations 16 (1991) 1223-1253) concerning solution sequences for the "regular" Lionville equation, where the Dirac measures are replaced by L-P(Omega)-data p > 1. In some particular case, we also derive a mass-quantization principle in the same spirit of Li-Shafrir (Indiana Univ. Math. J 43 (1994) 1255-1270). Our analysis was motivated by the study of the Bogornol'nyi equations arising in several self-dual gauge field theories of interest in theoretical physics, such as the Chern-Simons theory ("Self-dual Chern-Simons Theories," Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin, 1995) and the Electroweak theory ("Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions," World Scientific, Singapore). (C) 2002 Elsevier Scierice (USA).