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

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.