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


1. a subsequence $w\_{n\_k}$ is bounded in $L\_\mathrm{loc}^{\infty}(\Omega)$
2. $w\_n$ 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](https://doi.org/10.1080/03605309108820797)", Communications in Partial Differential Equations 16, No. 8-9, 1223-1253 (1991), [MR1132783](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1132783), [Zbl 0746.35006](https://zbmath.org/?q=an%3A0746.35006).



A detail in one step in a theorem from a paper of Brezis and Merle
This follows from the mean value theorem. Assume that (up to a subsequence) $w\_n(x\_n) \geq -B$ with $(x\_n) \in K$ (a compact subset of $\Omega$). If $x\_n \to x\_0 \in K$ and $B(x\_n,r) \in \Omega$ for every $n$, then $\int\_{B(x\_n,r)} w\_n \geq -B$ and then $$\int\_{B(x\_n,r)} w\_n^- \leq B+Ar^N, \quad \int\_{B(x\_n,r)} |w\_n| \leq B+2Ar^N$$ (here $w\_n^+ \leq A$) and $\int\_{B(x\_0,\frac r2)} |w\_n| \leq B+2Ar^N$ . The mean value property again yields $|w\_n(x)| \leq 4^N (Br^{-N}+2A)$ if $x \in B(x\_0, r/4)$.
Let $G$ be a connected compact set contained in $\Omega$ and containing $x\_0$ and $E$ the points $x \in G$ having a neighborhood where $(w\_n)$ is bounded. Then $x\_0 \in E$ and $E$ is open in $G$. Let $(x\_k) \subset E$ converge to $z \in G$. Then $|w\_n(x\_k)| \leq B\_k$ and, as above, $|w\_n|$ is uniformly bounded in a neighborhood of $x\_k$ which depends only on the distance of $x\_k$ from $\partial \Omega$. If $k$ is sufficiently large, this neighborhood contains $z$, hence $E$ is closed in $G$ and $E=G$. Now, it suffices to cover $G$ with a finite number open sets where $(w\_n)$ is bounded.


