does exists a Liouville theorem in this case?

Question

Let $u \in H^1 (\Omega)$ ($\Omega$ a smooth bounded domain in $R^n$) with $u $ bounded and $\Delta u = 0$ in $\Omega $ in the weak sense. Then u is constant?

I am trying to find in the internet some Liouville type theorem, but I am not finding anything that helps..

Someone could give me a help with the question?

thanks in advance!

Answer 1

No. $f(x)=x$ is harmonic and bounded on $(0,1)$ but not constant. If it is harmonic on $\mathbb{R}^n$ and bounded then it is constant, however.