Is there a solution to the following problem?
$-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$.
where $\Omega$ is bounded.

Try it on an interval first?
– Otis ChodoshMay 30 '12 at 14:37

Look up "strong maximum principle". For example around page 33 or 34 of Gilbarg and Trudinger, Elliptic Partial Differential Equations of Second Order
– Willie WongMay 30 '12 at 14:50

Elliptic Partial Differential Equations of Second Order– Willie Wong May 30 '12 at 14:50