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In this paper, we classify all solutions of
$$\left\{ \begin{align}
& -\Delta u=0\,\,\,\,\,\,\,\,\,\text{in}\,\,\,\mathbb{R}_{+}^{2}, \\
& \frac{\partial u}{\partial t}=-c{{\left| x \right|}^{\beta }}{{e}^{u}}\,\,\,\text{on}\,\,\partial \mathbb{R}_{+}^{2}\backslash \left\{ 0 \right\}, \\
\end{align} \right.$$
with the finite conditions
$${{\int }_{\partial \mathbb{R}_{+}^{2}}}|x{{|}^{\beta }}{{e}^{u}}\,ds\,<\,C,\,\,\,\,\frac{\sup }{\mathbb{R}_{+}^{2}}\,u(x)\,<\,C.$$
Here 
$c$ is a positive number and 
$\beta \,>\,-1$.
