# Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

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DOI: 10.1007/s002200200664

- Cite this article as:
- Bartolucci, D. & Tarantello, G. Commun. Math. Phys. (2002) 229: 3. doi:10.1007/s002200200664

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## Abstract

Motivated by the study of multivortices in the Electroweak Theory of Glashow–Salam–Weinberg [33], we obtain a concentration-compactness principle for the following class of mean field equations: \(\left( l \right)_\lambda - \Delta _g v = \lambda K\exp (v)/\int\limits_M {K\exp (v)d\tau _g - W}\) on *M*, where (*M*,*g*) is a compact 2-manifold without boundary, 0 < *a*≤*K*(*x*)≤*b*, *x*∈*M* and λ > 0. We take \(W = 4\pi \left( {\sum\limits_{i = 1}^m {\alpha _i \delta _{p_i } - \psi } } \right)\) with α_{i}> 0, δ_{p}_{i} the Dirac measure with pole at point *p*_{i}∈*M*, *i*= 1,…,*m* and ψ∈*L*^{∞}(*M*) satisfying the necessary integrability condition for the solvability of (1)_{λ}. We provide an accurate analysis for solution sequences of (1)_{λ}, which admit a “blow up” point at a pole *p*_{i} of the Dirac measure, in the same spirit of the work of Brezis–Merle [11] and Li–Shafrir [35]. As a consequence, we are able to extend the work of Struwe–Tarantello [49] and Ding–Jost–Li–Wang [21] and derive necessary and sufficient conditions for the existence of periodic N-vortices in the Electroweak Theory. Our result is sharp for *N*= 1, 2, 3, 4 and was motivated by the work of Spruck–Yang [46], who established an analogous sharp result for *N*= 1, 2.