On the mass of the exterior blow-up points

Abstract

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

arXiv commentarXiv admin note: substantial text overlap with arXiv:1309.1582, arXiv:1302.0657