Let $\Omega \in \mathbb{R}^n$ be a bounded domain, $f : \mathbb{R} \rightarrow \mathbb{R}$ continuous, $u\in C^2(\Omega)\cap C^0(\Omega)$ solution of
$\Delta u(x) = f(u(x))$ for all $x\in \Omega$
$u(x)=0$ for all $x\in \partial\Omega$
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I want to show now that:
a) if $0 \leq f(v)$ with $v\in \mathbb{R}$ then $u(x)<0$ for all $x\in \Omega$ or $u(x)=0$ for all $x\in \Omega$
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b) if $f$ monotonical increasing and $v\in C^2(\Omega)\cap C^0(\Omega)$
$\Delta v(x) = f(v(x))$ for all $x\in \Omega$
$0\leq v(x)$ for all $x\in \partial\Omega$
then it holds $u(x) \leq v(x)$ for all $x\in \Omega$
I have no clue how this should work out. Can anyone help?