## Double vortex condensates in the Chern-Simons-Higgs theory.(English)Zbl 0951.58030

Summary: For a selfdual model introduced by J. Hong, Y. Kim and P. Pac and R. Jackiw and E. Weinberg we studye the existence of double vortex-condensates “bifurcating” from the symmetric vacuum state as the Chern-Simons coupling parameter $$k$$ tends to zero. Surprisingly, we show a connection between the asymptotic behavior of the given double vortex as $$k\to 0^+$$ with the existence of extremal functions for a Sobolev inequality of the Moser-Trudinger type on the flat 2-torus. 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.

### MSC:

 58J90 Applications of PDEs on manifolds 81T13 Yang-Mills and other gauge theories in quantum field theory 58E30 Variational principles in infinite-dimensional spaces

### Keywords:

Chern-Simons-Higgs theory; double vortex-condensates
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