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Bubbling Solutions for Relativistic Abelian Chern-Simons Model on a Torus

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Abstract

We prove the existence of bubbling solutions for the following Chern-Simons-Higgs equation:

$$ \Delta u +\frac{1}{\varepsilon^2} e^u(1-e^u) =4\pi \sum_{j=1}^N \delta_{p_j},\quad {\rm in} \, \Omega, $$

where Ω is a torus. We show that if N > 4 and p 1 ≠ p j , j = 2, . . . , N, then for small ε > 0, the above problem has a solution u ε , and as ε → 0, u ε blows up at the vertex point p 1, and satisfies

$$ \frac{1}{\varepsilon^2} e^u(1-e^u)\to 4\pi N \delta_{p_1}. $$

This is the first result for the existence of a solution which blows up at a vertex point.

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Authors and Affiliations

  1. Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, 106, Taiwan

    Chang-Shou Lin

  2. Department of Mathematics, The University of New England, Armidale, NSW, 2351, Australia

    Shusen Yan

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  1. Chang-Shou Lin
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  2. Shusen Yan
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Correspondence to Chang-Shou Lin.

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Communicated by A. Kupiainen

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Lin, CS., Yan, S. Bubbling Solutions for Relativistic Abelian Chern-Simons Model on a Torus. Commun. Math. Phys. 297, 733–758 (2010). https://doi.org/10.1007/s00220-010-1056-1

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  • DOI: https://doi.org/10.1007/s00220-010-1056-1

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