Let u be a harmonic non negative function in $\Omega=B(0,1)\setminus\{0\} \in\mathbb R^n$. I would like to show that there exists a constant c, depending just on $n$ such that $$ \max_{\partial B(0,r)} u \leq c \min_{\partial B(0,r)} u $$ for every $0<r\leq \frac{1}{2}$.
My attempt is to use the Harnack's inequality which implies that for all the balls $B(x_0,r) $ s.t. $B(x_0,4r)\subset \Omega$, $x_0 \in \Omega$, then $$ \max_{ \overline{B(x_0,r)}} u \leq 3^n \min_{\overline{ B(0,r)}} u $$ Then I would use the maximum principle to conclude.
The problem is that for every $r$ the ball $B(0,1)$ does not belong to $\Omega$, hence I am not able to apply the Harnack's inequality. How could I avoid this problem?