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

We prove the existence of bubbling solutions for the following Chern-Simons-Higgs equation: Delta u + 1/epsilon(2)e(u)(1 - e(u)) = 4 pi Sigma(N)(j=1) delta(pj), in Omega, where Omega is a torus. We show that if N > 4 and p(1) not equal p(j), j = 2, ... , N, then for small epsilon > 0, the above problem has a solution u(epsilon), and as epsilon -> 0, u(epsilon) blows up at the vertex point p(1), and satisfies 1/epsilon(2)e(u)(1 - e(u)) -> 4 pi N delta(p1). This is the first result for the existence of a solution which blows up at a vertex point.