Let $u \in C^{\infty}(\mathbb{R}^3)$ be a harmonic function on $\mathbb{R}^3$: $\Delta u(x) = 0$ for all $x \in \mathbb{R}^3$.
Assume that $|u(x)| \leq \sqrt{|x|}$ holds for all $x$, where $x = (x_1,x_2,x_3)$. Show that $u(x) = 0$ for all $x \in \mathbb{R}^3$.
I want to apply Harnack Inequality to $u$ on $B_R(0)$ because then I can get something like $$u(x) \leq \frac{R(R+|x|)}{(R - |x|)^2}u(0) \leq \frac{R(R+|x|)}{(R - |x|)^2}\sqrt{|0|} \leq 0.$$ However this only applies when $u$ is non-negative on the ball, which is not something I am given.
Thanks.