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

Abstract

where pj ε Ω, αj > 0 and δpj denotes the Dirac measure with pole at point pj, J = 1,⋯,m. Our result extends Brezis-Merle's theorem (Comm. Partial Differential Equations 16 (1991) 1223-1253) concerning solution sequences for the "regular" Liouville equation, where the Dirac measures are replaced by Lp(Ω)-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 Bogomol'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). © 2002 Elsevier Science (USA).