On the mass of the exterior blow-up points

Abstract

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: $\int_{\Omega}V_i e^{u_i} dy \leq 40\pi-\epsilon , \,\, \epsilon >0$.