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.
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I $\Omega$ is a bounded domain, the answer is NO. Because of $$\int_{\Omega}\Delta v dx=\int_{\partial\Omega}\frac{\partial v}{\partial\nu}ds.$$ 

