  We consider the following problem on open set $\Omega$ of ${\mathbb R}^2$:
$$\left \{ \begin {split} -\Delta u_i & = V_i e^{u_i} \,\, &\text{in} \,\,
&\Omega \subset {\mathbb R}^2, \\ u_i & = 0 \,\, & \text{in} \,\, &\partial
\Omega.\end {split}\right.$$ We assume that $ \int_{\Omega} e^{u_i} dy \leq C$,
and $ 0 \leq V_i \leq b < + \infty$.
  On the other hand, if we assume that $V_i$ $s-$holderian with $1/2< s \leq
1$, then each exterior blow-up point is simple. As application, we have a
compactness result for the case when: $\int_{\Omega}V_i e^{u_i} dy \leq
40\pi-\epsilon , \,\, \epsilon >0$.
