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

TL;DR

This paper investigates the asymptotic behavior of positive, axially symmetric solutions to a degenerate elliptic equation with a nonlinear boundary condition in the upper half-space, focusing on the case where the degeneracy parameter is negative.

Contribution

It provides qualitative properties and asymptotic expansions of solutions for the degenerate elliptic problem with nonlinear boundary conditions, specifically for the case when the degeneracy parameter is in (-1,0).

Findings

Established asymptotic expansion of solutions.

Proved qualitative symmetry properties.

Analyzed solutions under specific parameter conditions.

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 \nu^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 $\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. We obtain some qualitative properties of positive axially symmetric solutions in $n\geq3$ for the case $a\in (-1,0)$ under the condition $q\geq\frac{n-a}{n+a-2}$. In particular, we establish the asymptotic expansion of positive axially symmetric solutions.