I am currently practising some past year exams questions on intro to PDE in my school and I have problem doing the following:
Let $\Omega$ be a bounded domain and let $u$ satisfies: $-\Delta u+ c(x)u=f(x)$ for $x\in \Omega$ and $u=0$ on $\partial\Omega$
Prove that:
If $c(x)\geq c>0$ with a positive constant $c$, then $\max_{\Omega} |u(x)| \leq \frac{1}{c} \sup_{\Omega} |f(x)|$.
If $0\leq c(x)\leq d<\infty$ with a constant $d$, then $\max_{\Omega} |u(x)| \leq M\, \sup_{\Omega} |f(x)|$, where $M$ depends only on $d$.
I tried to multiply the equation by $u$ and integrate it, so I get something like $\int_\Omega c(x) u-f(x) u^2\;dx <0$, but I don't think I am getting closer to the ans.
Any help would be greatly appreciated. Thanks!