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Classification of Solutions for Harmonic Functions With Neumann Boundary Value

  • Tao Zhang (a1) and Chunqin Zhou (a2)


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 $\beta \,>\,-1$ .



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