Classification of Solutions for Harmonic Functions With Neumann Boundary Value

Zhang, Tao ; Zhou, Chunqin

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 438-448

Résumé

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

Détail

DOI : 10.4153/CMB-2017-037-1

Mots-clés : 35A05, 35J65, Neumann problem, singular coeffcient, classification of solutions