$u$ harmonic and $u^3$ harmonic then $u$ constant

Question

Let $u(x,y)$ be a harmonic function defined in a connected open subset of $\Bbb R^2$. Does $u^3(x,y)=(u(x,y))^3$ harmonic implies $u$ is constant? It is easily shown as true, when I replace $3$ with $2$, so I am wondering that this holds for all $n>2$.

Answer 1

$\partial_x^2(u^k)=\partial_x(ku^{k-1}\partial_xu)=ku^{k-1}\partial_x^2u+k(k-1)u^{k-2}(\partial_xu)^2$.

So $\Delta(u^k)=ku^{k-1}\Delta u+k(k-1)u^{k-2}|\nabla u|^2$. It follows that if $\Delta(u^k)=0$ and $\Delta_u=0$, for some $k \geq 3$, then $u|\nabla u|=0$. Thus if on some open subset $|\nabla u| > 0$, $u$ vanishes on the set and we get a contradiction, thus $u$ is constant.