In paper [1] Brezis and Merle prove theorem 3 by using the following fact. Let $w_n=u_n-v_n$, $\Delta w_n=0$ on $\Omega$ (a bounded domain in $\mathbb{R^2}$) and $w_n^+$ is bounded in $L^{\infty}_\mathrm{loc}(\Omega)$. Then by Harnack's principle either

a subsequence $w_{n_k}$ is bounded in $L_\mathrm{loc}^{\infty}(\Omega)$

$w_n$ converges uniformly to $-\infty$ on compact subsets of $\Omega$.

Here $v_n$ is a solution of the boundary value problem $$ \begin{cases} -\Delta v_n=V_n e^{u_n} &\text{in }\Omega\\ \quad\: v_n=0 &\text{on }\partial\Omega \end{cases}$$ while $u_n$ solves $-\Delta u_n= V_n e^{u_n}$.

Since there is no boundary condition on $u_n$, I cannot say anything about the increasing or decreasing of the sequence $w_n$, while this would be required for the application of Harnack's principle as stated in most of the references I'aware of.

Any help would be appreciated: in particular I hope to get a hint regarding the use of the customary Harnack's principle in this case in order to prove the above claim.

**Reference**

[1] Haïm Brezis, Frank Merle, "Uniform estimates and blow up behavior of $-\Delta u=Ve^u$ in two dimensions", Communications in Partial Differential Equations 16, No. 8-9, 1223-1253 (1991), MR1132783, Zbl 0746.35006.