# Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

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 $$\sup |\Phi|, can we find a uniform bound C for any $$\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.