On the Ambjorn-Olesen electroweak condensates

We obtain sufficient conditions for the existence of the Ambjorn-Olesen [“On electroweak magnetism,” Nucl. Phys. B315, 606–614 (1989)] electroweak N-vortices in case N ≥ 1 and therefore generalize earlier results [D. Bartolucci and G. Tarantello, “Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory,” Commun. Math. Phys. 229, 3–47 (2002); J. Spruck and Y. Yang, “On multivortices in the electroweak theory I: Existence of periodic solutions,” ibid. 144, 1–16 (1992)] which handled the cases N ∈ {1, 2, 3, 4}. The variational argument provided here has its own independent interest as it generalizes the one adopted by Ding et al. [“Existence results for mean field equations,” Ann. Inst. Henri Poincare, Anal. Non Lineaire 16, 653–666 (1999)] to obtain solutions for Liouville-type equations on closed 2-manifolds. In fact, we obtain at once a second proof of the existence of supercritical conformal metrics on surfaces with conical singularities and prescribed Gaussian curvature recently established by Bartolucci, De Marchis and Malchiodi [Int. Math. Res. Not. 24, 5625–5643 (2011)].