Recently, I read this paper about $L^p$ Minkowski problem. In page $84$, it said that
From $u_0>0$, one can see that there exist constants $m$ and $M$, s.t. $$0<m\leq\frac{1}{u_k}\leq M.$$ Here $u_k \rightharpoonup u_0$ in $H^1(S)$.
I wonder how it works. I think $m$ and $M$ are related to $u_0$, but if $u_0$ is bounded is unknown. So how we find the constants $m$ and $M$? I wonder the details of this step. Thanks in advance!
$S$ is the sphere $S^1$, so it is compact. Also $H^1(S)$ is compactly embedded in $C(S)$ (see equation above (17)). $u_0>0$ implies that $u_0\ge 2m$ for some $m>0$ as $u_0$ is continuous on the compact set $S$.