If $u \geq 0$ is harmonic in $\Omega \subset \mathbb{R}^n$ open and connected, then either $u \equiv 0$ or $u > 0$ in every $\Omega$

Question

I need to prove:

If $u \geq 0$ is harmonic in $\Omega \subset \mathbb{R}^n$ open and connected, then either $u \equiv 0$ or $u > 0$ in every $\Omega$.

I sought to use Harnack's principle or inequality. Help me, I need more details...

Answer 1

For any $x\in\Omega$, $u(x)$ is the average value of $u$ on any ball $B$ centered at $x$. Were we to have $u(x) = 0$ for some $x\in \Omega$, then $u$ would be 0 on any ball $B$ centered at $x$ with $B\subseteq \Omega$. Now use the fact that $\Omega$ is connected.