I am strugging on problem 3.2 in Gilbarg-Trudinger, which says that
if $L=a^{ij}(x)D_{ij}+b^{i}(x)D_{i}+c(x)$ is an elliptic operator in a bounded domain $\Omega \subset \mathbb{R}^{n}$ with $c<0$, and $u\in C^{2}(\Omega)\cap > C^(\overline\Omega)$ satisfies $Lu=f$ in $\Omega$, then we have $\sup_{\Omega}|u| \le\sup_{\partial \Omega}|u|+\sup_{\Omega}|\frac{f}{c}|$
In previous part of this chapter, we have actually shown that for the case $c\le0$, we have $\sup_{\Omega}|u| \le\sup_{\partial \Omega}|u|+C\sup_{\Omega}|\frac{f}{\lambda}|$, where $\lambda(x)$ is the minimum eigenvalue of $[a^{ij}(x)]$ and $C$ is a constant depending only on $diam(\Omega)$ and $\beta=\sup \frac{|\mathbf{b}|}{\lambda} (<\infty)$.
I have no idea why the bound can be independent of $diam(\Omega)$ and $\beta$, can anybody give me some idea?