The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula

For a bounded domain Ω 2, we establish a concentration-compactness result for the following class of “singular” Liouville equations: where pj Ω, αj>0 and δpj denotes the Dirac measure with pole at point pj, j=1,…,m. Our result extends Brezis–Merleʹs theorem (Comm. Partial Differential Equations16 (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 Bogomolʹnyi 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).