Asymptotic behavior of positive solutions to a degenerate elliptic equation in the upper half space with a nonlinear boundary condition

Abstract

We consider positive solutions of the problem \begin{equation} \left\{\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=0\qquad \mbox{in}\;\;\mathbb{R}_+^n,\\ \frac{\partial u}{\partial ν^a}=u^{q} \qquad \mbox{on}\;\;\partial \mathbb{R}_+^n,\\ \end{array} \right. \end{equation} where $a\in (-1,0)\cup (0,1)$, $q>1$ and $\frac{\partial u}{\partial {\nu }^a}:=-{\lim }_{x_n\to 0^{+}}{x}_n^a\frac{\partial u}{\partial x_n}$. We obtain some qualitative properties of positive axially symmetric solutions in $n\ge 3$ for the case $a\in (-1,0)$ under the condition $q\ge \frac{n-a}{n+a-2}$. In particular, we establish the asymptotic expansion of positive axially symmetric solutions.

28 pages