Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \text { in } \Omega \\ \Phi &=0 \text { on } \partial \Omega \end{aligned}\right. $$ where $f(x, y)$ is smooth on $\bar{\Omega}$.
Then for any fixed $\sigma$, we have $\sup |\Phi|<C_{\sigma}$, can we find a uniform bound C for any $\sigma \in[0,1]$, the $C^{0}$ norm of the solution of $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \text { in } \Omega \\ \Phi &=0 \text { on } \partial \Omega \end{aligned}\right. $$ is bounded.
It follows by standard PDE regularity. You may check Theorem 8.16 in Trudinger's book. I think it is bounded by something like the $L^p$-norm of $\sigma f(x,y)$. Taking $\sigma = 1$, then I think the bound is uniform.