Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not successful. Thank you.
Let $u$ be a harmonic function in $\Bbb{R}^n$, $n\ge2$, and suppose $u(p)=0$ or some point $p$. Show that in any ball $B_r(p)$ there is another point $q\in B_r(p)$ different from $p$ where $u(q)=0$.
The mean value property tells us (in particular) that for any ball $B_r(p)$, $0=u(p)=\frac{1}{|B_r|}\int_{B_r(p)}u(y)$, WLOG assume that $u(x)>0$ in $B_r(p)\setminus\{p\}$, then $\frac{1}{|B_r|}\int_{B_{r(p)}}u(y)>0$, noting that $u$ is continuous, this contradicts the mean value property.
Note that this breaks down for $n=1$, take the example $f(x)=ax$, $a\in\Bbb R$.