Let $\Omega \subset \mathbb{R}^n,~n \geq3$ be a bounded domain with a smooth boundary. Does there exist any harmonic function $u \in C^{\infty}(\Omega)$ such that $u$ is bounded from below (or above), but not bounded from the other side?
A similar question is here, for $\mathbb{R}^2 = \mathbb{C}$ and $\Omega$ is the half-disk, which is different from this situation.