Counterexample to maximum principle of harmonic functions on unbounded domain

Question

The (weak) maximum principle for harmonic functions asserts that if $u$ is harmonic on the bounded set $\Omega \subset \mathbb R^n$, then $sup_\Omega u = sup_{\partial \Omega} u$. What would be a simple counterexample to this statement if $\Omega$ is unbounded (but has nonempty boundary, e.g. $|x| > 1$)?

Answer 1

The simplest counterexample is probably the function $u(x)=x_1$ on the half-space $\Omega = \{x\in\mathbb{R}^n:x_1>0\}$. It is equal to zero on the boundary, but is positive in the domain.