Article

Communications in Mathematical Physics

, Volume 297, Issue 3, pp 733-758

First online:

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

  • Chang-Shou LinAffiliated withDepartment of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University Email author 
  • , Shusen YanAffiliated withDepartment of Mathematics, The University of New England

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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 1p 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.