Communications in Mathematical Physics

, Volume 229, Issue 1, pp 3–47

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

  • D. Bartolucci
  • G. Tarantello

DOI: 10.1007/s002200200664

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


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 < aK(x)≤b, xM and λ > 0. We take \(W = 4\pi \left( {\sum\limits_{i = 1}^m {\alpha _i \delta _{p_i } - \psi } } \right)\) with αi> 0, δpi the Dirac measure with pole at point piM, 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 pi 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • D. Bartolucci
    • 1
  • G. Tarantello
    • 1
  1. 1.University of Rome “Tor Vergata”, Mathematics Departement, Via della Ricerca Scientifica, 00133 Rome, Italy. E-mail:; tarantel@axp.mat.uniroma2.itIT

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