For a selfdual model introduced by Hong-Kim-Pac  and Jackiw-Weinberg  we study the existence of double vortex-condensates "bifurcating" from the symmetric vacuum state as the Chern-Simons coupling parameter Ic tends to zero. Surprisingly, we show a connection between the asymptotic behavior of the given double vortex as k --> 0(+) with the existence of extremal functions for a Sobolev inequality of the Moser-Trudinger's type on the flat 2-torus (,  and ). In fact, our construction yields to a "best" minimizing sequence for the (non-coercive) associated extremal problem, in the sense that, the infimum is attained if and only if the given minimizing sequence admits a convergent subsequence.
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