Abstract view
Classification of solutions for harmonic functions with Neumann boundary value


Tao Zhang,
Department of Mathematics, Shanghai Jiaotong University, 200240, Shanghai, China
Chunqin Zhou,
Department of Mathematics and MOELSC, Shanghai Jiaotong University, 200240, Shanghai, China
Abstract
In this paper, we classify all solutions of
\[
\left\{
\begin{array}{rcll}
\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+},
\\
\dfrac{\partial u}{\partial t}&=&cx^{\beta}e^{u} \quad
&\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\},
\\
\end{array}
\right.
\]
with the finite conditions
\[
\int_{\partial \mathbb{R}^{2}_{+}}x^{\beta}e^{u}ds \lt C,
\qquad
\sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C.
\]
Here, $c$ is a positive number and $\beta \gt 1$.