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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 MOE-LSC, Shanghai Jiaotong University, 200240, Shanghai, China
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Abstract

In this paper, we classify all solutions of \[ \left\{ \begin{array}{rcll} -\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+}, \\ \dfrac{\partial u}{\partial t}&=&-c|x|^{\beta}e^{u} \quad &\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\}, \\ \end{array} \right. \] 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 $\beta \gt -1$.
Keywords: Neumann problem, singular coefficient, classification of solutions Neumann problem, singular coefficient, classification of solutions
MSC Classifications: 35A05, 35J65 show english descriptions General existence and uniqueness theorems
Nonlinear boundary value problems for linear elliptic equations
35A05 - General existence and uniqueness theorems
35J65 - Nonlinear boundary value problems for linear elliptic equations
 

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