Bubbling solutions for a skew-symmetric Chern-Simons system in a torus

Abstract

We establish the existence of bubbling solutions for the following skew-symmetric Chern Simons system Delta u(1) + 1/epsilon(2) e(u2) (1 - e(u1)) = 4 pi Sigma N-1 i=1 delta p(1/i) { Delta u(2) + 1/epsilon(2) e(u1) (1 - e(u2)) = 4 pi Sigma N-2 i=1 delta p(2/i) over a parallelogram Omega with doubly periodic boundary condition, where epsilon > 0 is a coupling parameter, and delta(p) denotes the Dirac measure concentrated at p. We obtain that if (N-1 - 1)(N-2 - 1) > 1, there exists an epsilon(o) > 0 such that, for any epsilon is an element of(0, epsilon(o)), the above system admits a solution (u(1),(epsilon), u(2),(epsilon)) satisfying u1,(epsilon) and u(2),(epsilon) blow up simultaneously at the point p*, and 1/epsilon(2) e(uj,k) (1 - e(ui,epsilon)) -> 4 pi N-i delta(p*) , 1 <= i, j <= 2 , i not equal j as epsilon -> 0, where the location of the point p* defined by (1.12) satisfies the condition (1.13). (C) 2017 Elsevier: Inc. All rights reserved.