I am studying from the book of Qing Han- A Basic Course in Partial Differential Equations
Can't find any clue how to solve: Exercise 4.8. 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 $$ \begin{array}{ll} \lim _{|x| \rightarrow \infty} \frac{u(x)}{\ln |x|}=0 & \text { for } n=2, \\ \lim _{|x| \rightarrow \infty} u(x)=0 & \text { for } n \geq 3 \end{array} $$
Should I look for symmetric solutions? Anything connected to the fundamental solution?
There is some connection with the fundamental solution. Hint: the Kelvin tramsform and a removable singularity theorem for harmonic functions.