Let $n\geq3$, $B_{1}\left( 0\right) \subset\mathbb{R}^{n}$ and $u\in C\left( \overline{B_{1}\left( 0\right) }\backslash\left\{ 0\right\} \right) $ be harmonic. Suppose $u\left( x\right) =0$ on $\partial B_{1}\left( 0\right) $ and $$ \lim_{x\rightarrow0}\left\vert x\right\vert ^{n-2}u\left( x\right) =0. $$ Show that $u\equiv0$ in $B_{1}\left( 0\right) $. (Hint: Compare $u$ with some radially symmetric harmonic function.)
I am unable to solve the problem. In fact, I am not getting any direction on where to start. The function is not harmonic in the unit ball, it is harmonic on the punctured disc. So, I cannot use the mean value theorem. The function is not even defined at $0$. If someone can even give me a starting point with some more direction with hints. I will be grateful and I will try to fill up the details.