Double vortex condensates in the Chern-Simons-Higgs theory

For a selfdual model introduced by Hong-Kim-Pac [18] and Jackiw-Weinberg [1 9] we study the existence of double vortex-condensates "bifurcating" from t he symmetric vacuum state as the Chern-Simons coupling parameter Ic tends t o zero. Surprisingly, we show a connection between the asymptotic behavior of the given double vortex as k --> 0(+) with the existence of extremal fun ctions for a Sobolev inequality of the Moser-Trudinger's type on the flat 2 -torus ([22], [1] and [15]). In fact, our construction yields to a "best" m inimizing sequence for the (non-coercive) associated extremal problem, in t he sense that, the infimum is attained if and only if the given minimizing sequence admits a convergent subsequence.