Singular limits for Liouville-type equations on the flat two-torus (Q2636879)

This paper is concerned with the problem \[ -\Delta u=\varepsilon^2 e^u-\frac{1}{|\Omega|}\int_{\Omega} \varepsilon^2 e^u +\frac{4\pi N}{|\Omega|}-4\pi N\delta_P\quad\text{ in }\Omega,\quad\int_{\Omega} u=0, \] where \(\Omega\subset{\mathbb R}^2\) is a flat torus, \(\varepsilon>0\) and \(p\in \Omega\). The author proves that for \(1\leq m<N+1\) then there exists a family of solutions \(\{u_\varepsilon\}\) such that \(\varepsilon^2 e^{u_\varepsilon}\to 8\pi \sum_{i=1}^m \delta_{p_i}\) in the measure sense as \(\varepsilon\to 0\), where \(q_1,q_2,\dots,q_m\in \Omega\setminus\{p\}\) are distinct points.