Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1} -\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right), $$ on flat torus $\Omega$. They tried to use the following celebrated lemma by Brezis and Merle
(Brezis-Merle). - Let $D$ be a bounded domain in $\mathbb{R}^2$ and $\left\{w_n\right\}$ be a sequence satisfying: $$ -\Delta w_n=V_n(x) e^{w_n} \quad \text { on } D $$ with $0 \leqslant V_n(x) \leqslant b_1$ on D. Also suppose that $\int_D e^{w_n} \leqslant b_2$. Then $\left\{w_n\right\}$ admits a subsequence $\left\{w_{n_k}\right\}$ satisfying one of the following:
i) $\left\{w_{n_k}\right\}$ is uniformly locally bounded in $D$;
ii) for any compact set $K \subset D$, there holds $$ \sup _K w_{n_k} \rightarrow-\infty \quad \text { as } k \rightarrow+\infty ; $$
iii) there exists $S=\left\{a_1, \ldots, a_p\right\} \subset D$ (blow up set) and a sequence $\left\{x_{n_k}^i\right\} \subset D$ such that, as $k \rightarrow \infty, x_{n_k}^i \rightarrow a_i, w_{n_k}\left(x_{n_k}^i\right) \rightarrow \infty, i=1, \ldots, p$. Moreover, for any compact set $K \subset D \backslash S$ we have, $\sup w_{n_k} \rightarrow-\infty$ as $k \rightarrow \infty$. (Li-Shafrir): In addition, if $V_n \rightarrow V$ in $C^0(\bar{\Omega})$, then $$ V_{n_k} e^{w_{n_k}} \rightarrow \sum_{i=1}^p 8 \pi m_i \delta_{x=a_i} $$ in the sense of measures, with $m_i \in \mathbb{N}$ and $\delta_{x=a_i}$ the Dirac distribution supported in $\left\{a_i\right\}, i=1, \ldots, p$.
Let $\lambda_n \rightarrow \lambda$ and let $u_n \in E$ be a solution for $(1)_{\lambda_n}$ . They deal the equation in the following way: set $$ u_n(0)=\sup _{\Omega} u_n, $$ and let $$\tag{2} w_n(x)=u_n(x)-\ln \left(\int_{\Omega} e^{u_n} d x\right)-\frac{\lambda_n}{4}|x|^2 \text {, } $$ with $$ \int_{\Omega} e^{w_n} d x \leqslant 1 . $$
Thus, the hypotheses of Lemma by Brezis-Merle are satisfied for $w_n$ with $V_n=\lambda_n e^{\left(\lambda_n / 4\right)|x|^2} \leqslant$ $\lambda_n e^{\lambda_n}, b_2=1$. (If we set $\lambda \in (8 \pi, 4 \pi^2)$, then (iii) can not occur.)
But there is a problem, what the definition of the variable of $w_n$ in (2)?, to satisfy the Brezis-Merle lemma, it should be a bounded domain in $\mathbb{R}^2$, but the variable on $u_n$ is on torus, how did they mixed these two things together?
Or this $| \cdot |$ is a geodesic distance? If so, they then estimated $\sup _{B_{1 / 2}(0)} w_n$, is this $B_{1 / 2}(0)$ a geodesic ball?
Or does this mean that they just simply treated it as an equation on a square with opposite sites topologically identified? In fact, in the beginning of the paper, they said Let $\Omega$ be the 2 -dimensional torus, with fundamental cell domain: $[-1 / 2,1 / 2] \times$ $[-1 / 2,1 / 2]$. Or, equivalently, solutions of $$ -\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-1\right) \quad \text { on } \Omega $$ on $\mathbb{R}^2$ of period 1 in each variable.
