Harmonic function in $\mathbb{R}^n$ is not one-to-one, for $n\geq 2.$

Question

Le $u:\mathbb{R}^n\to\mathbb{R}$ a harmonic function. Prove that if $n\geq2$ then every $y\in Im\{u\}$ is attained infinite times, but it's not true for $n=1$.

I no have idea to start, someone has a hint?

Thanks.

Answer 1

For example if $u(y) = c$, then for each $r$ there is $x$ so that $u(x) = c$ and $|x-y| = r$. This is because $c = u(y)$ is the average of $u(x)$ on $\partial B(y, r)$, so it is impossible that all values on this $\partial B(y, r)$ are $>c$ or $<c$.

Since there are infinitely many choices of $r>0$, the value $c$ is attained infinitely many times.

(Taken from John's comments)