Existence of Bubbling Solutions for Chern-Simons Model on a Torus

We study the existence of bubbling solutions for the the following Chern-Simons-Higgs equation: Delta u + 1/epsilon(2) e(u) (1 - e(u)) = 4 pi Sigma(2k)(i=1) delta p(i), in Omega, where Omega is a torus. If k = 1, for any critical point q of the associated sum of the Green functions, we introduce a quantity D(q) (see (1.11) below). We show that for any non-degenerate critical point q with D(q) < 0, the above problem has a solution u (epsilon) satisfying that epsilon -> 0, u (epsilon) blows up at q. The calculations in this paper also show that, if a sequence of solutions u (epsilon) blows up at q as epsilon -> 0, then q must be a critical point of the associated sum of the Green functions, and . So, the condition D(q) < 0 is almost necessary to obtain our result. We also construct solutions with k bubbles for .