Maximum of elliptic PDE

Question

Let $\Omega =(0,1)^2$ and \begin{align} -div(p\nabla u)(x,y)=f(x,y) \text{ for } (x,y)\in \Omega\\ \end{align} for $p\in C^1$ and $u\in C^2(\Omega)\cap C(\overline{\Omega})$. If $f<0$ and $p\geq 0$ I know that $f$ takes its maximum at $\partial \Omega$ which can easily be shown by a prove by contradiction. But is it still necessary that $p\geq 0$?

Answer 1

Here is a contradiction for $p<0$. Choose $p\equiv-1$ and $f\equiv-1$ on the domain $\Omega$. Then, we want to find a solution to $\Delta u =-1$ on $\Omega$. It is easy to check that $u =\frac{1}{4}\left( -(x-1/2)^2 -(y-1/2)^2\right)$ is a solution whose maximum is not at the boundary.