If I consider a Liouville equations in the unit disk $D \subset \mathbb{R}^2$ with singularity at $x=0$, $$\Delta u= e^{2u}$$ If I define the order of $u$ at origin is defined to be $$\lim_{r \rightarrow 0}\frac{\max\limits_{|x|=r} u}{\log(1/r)}=\alpha$$ It is known that $u$ has the expression \begin{array}{lcl} u = - \alpha \log | x | + w_i, &&\text{if $\alpha < 1$;}\\ u = - 2 \log | x | - 2 \log \log \left( | x |^{-1} \right) + \widetilde{w_i}, &&\text{if $\alpha = 1$.} \end{array} Here the remainder function $w_i$ and $\widetilde{w_i}$ are continuous functions near origin.

(see: http://www.sciencedirect.com/science/article/pii/S0022247X83712588)

But what happen if $\alpha >1$?

If I consider a Liouville equations in the unit disk $D \setminus\{0\} \subset \mathbb{R}^2$ with singularity at $x=0$, $$\Delta u= e^{2u}$$ If I define the order of $u$ at origin is defined to be $$\lim_{r \rightarrow 0}\frac{\max\limits_{|x|=r} u}{\log(1/r)}=\alpha$$ It is known that $u$ has the expression \begin{array}{lcl} u = - \alpha \log | x | + w_i, &&\text{if $\alpha < 1$;}\\ u = - 2 \log | x | - 2 \log \log \left( | x |^{-1} \right) + \widetilde{w_i}, &&\text{if $\alpha = 1$.} \end{array} Here the remainder function $w_i$ and $\widetilde{w_i}$ are continuous functions near origin.

(see: http://www.sciencedirect.com/science/article/pii/S0022247X83712588)

But what happen if $\alpha >1$?