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

We establish the existence of bubbling solutions for the following skew-symmetric Chern-Simons system{Δu1+1ε2eu2(1−eu1)=4π∑i=1N1δpi1Δu2+1ε2eu1(1−eu2)=4π∑i=1N2δpi2 over a parallelogram Ω with doubly periodic boundary condition, where ε>0 is a coupling parameter, and δp denotes the Dirac measure concentrated at p. We obtain that if (N1−1)(N2−1)>1, there exists an ε0>0 such that, for any ε∈(0,ε0), the above system admits a solution (u1,ε,u2,ε) satisfying u1,ε and u2,ε blow up simultaneously at the point p⁎, and1ε2euj,ε(1−eui,ε)→4πNiδp⁎,1≤i,j≤2,i≠j as ε→0, where the location of the point p⁎ defined by (1.12) satisfies the condition (1.13).