Construction of blowup solutions for Liouville systems

We study the following Liouville system defined on a flat torus \begin{equation} \left{ \begin{array}{lr} -Δu_i=\sum_{j=1}^n a_{ij}ρ_j\Big(\frac{h_j e^{u_j}}{\int_Ωh_j e^{u_j}}-1\Big),\nonumber \ u_j\in H_{per}^1(Ω)\mbox{ for }i\in I={1,\cdots,n}\nonumber, \end{array} \right. \end{equation} where $h_j\in C^3(\mathrm{\Omega })$, $h_j>0$, ${\rho }_j>0$ and $u=(u_1,\mathrm{.}\mathrm{.},u_n)$ is doubly periodic on $\mathrm{\partial }\mathrm{\Omega }$. The matrix $A=(a_{ij}{)}_{n\times n}$ satisfies certain properties. One central problem about Liouville systems is whether multi-bubble solutions do exist. In this work we present a comprehensive construction of multi-bubble solutions in the most general setting.