equation $\Delta u=0$ should be satisfied in the disk: $\{r<a\}$ rather than in the disk without its center $\{0<r<a\}$. There is a maximum principle for Laplace, which means that $u$ must be bounded at $0$,

\begin{equation}

\max |u|<\infty.\tag{*}

\end{equation}

This eliminates $r^\nu$ with $\nu<0$ and $\ln(r)$.

Actually there is much stronger statement: $u$ is $C^\infty$ and even real-analytic (could be decomposed into Taylor series by $x,y$ of non-zero radius of convergence). This eliminates $r^\nu$ with positive but non-integer $\nu$.

However, if we consider a sector $\{r<a, \theta_1<\theta <\theta_2\}$ we need to request (*) explicitly. Ditto in unbounded domains.