Let $u$ be a $C^{2}$ -solution of $$ \begin{aligned} \Delta u=0 & \text { in } \mathbb{R}^{n} \backslash B_{R} \\ u=0 & \text { on } \partial B_{R} \end{aligned} $$ Prove that $u \equiv 0$ if $$ \begin{array}{ll} \lim _{|x| \rightarrow \infty} \frac{u(x)}{\ln |x|}=0 & \text { for } n=2, \\ \lim _{|x| \rightarrow \infty} u(x)=0 & \text { for } n \geq 3 \end{array} $$
I've tried to use Kelvin's transform https://en.wikipedia.org/wiki/Kelvin_transform in order to get compect domain. Will it allow me to use the Mean Value theorem? Maximum principle?