Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory
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Motivated by the study of multivortices in the Electroweak Theory of Glashow–Salam–Weinberg , we obtain a concentration-compactness principle for the following class of mean field equations: on M, where (M,g) is a compact 2-manifold without boundary, 0 < a≤K(x)≤b, x∈M and λ > 0. We take 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  and Li–Shafrir . As a consequence, we are able to extend the work of Struwe–Tarantello  and Ding–Jost–Li–Wang  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 , who established an analogous sharp result for N= 1, 2.
- Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory
Communications in Mathematical Physics
Volume 229, Issue 1 , pp 3-47
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