Find all Harmonic functions $u:\Bbb{R}^n \to \Bbb{R}$ st $|u(x)|\le C|x|^m$, for all $|x|\ge 1$ where $C$ is constant and $m\in (0,2)$.
I tried to use the same argument as used in the proof of Liouville's theorem but that isn't working here since $|x|\ge 1$. And I believe that these functions will be constant.
Recall one of the fundamental estimates for harmonic functions $$ | \nabla u(x)|\leq C(n) R^{-1}\sup_{B_R(x)}|u|. $$ Taking $R=|x|/2$ we get $$ |\nabla u(x)|\leq C|x|^{m-1}, \qquad \text{ for }|x|>2. $$ If $m<1$ this implies that $\nabla u$ is a bounded harmonic function decaying at $\infty$. Therefore $\nabla u=0$ and so $u$ is constant. If $m\geq 1$, then applying the case $m<1$ to $v_i=\partial_i u$ we conclude that $v_i=c_i$ constant. Therefore $u$ is a linear function in this case.
