Does there exist $u$ that is not a constant function such that $u^2$ is harmonic?

Question

As the title says, does there exist $u$ : $\mathbb R^2 \to \mathbb R$ that is not a constant function such that $u^2$ is harmonic?
What if $u$ is continuous? What if $u$ is harmonic?

Answer 1

1. If $u$ is any function, take $u$ to be 1 anywhere except in the origin, where it is -1. Then $u^2$ is constant, hence harmonic.
2. If $u$ is continuous, then we know there's a complex function $f$ such that $Re(f)=u^2$. Then we get $Re(f) \geq 0$, and we can take $|e^{-f}|=|e^{-u^2}|$ which is bounded, hence by Liouville constant, and then f is constant and $u^2$ is constant (little Picard theorem also works).
3. If $u$ is harmonic then it is continuous, so it is compatible with case 2.