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arXiv 1504.04095math.AP
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上半空间中一类退化椭圆型方程非线性边值问题正解的渐近性

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

Zhuoran Du

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28 pages
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摘要：

我们考虑问题\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}的正解，其中$a\in (-1,0)\cup(0,1)$，$q>1$和$\frac{\partial u}{\partial ν^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$ .在条件$q\geq\frac{n-a}{n+a-2}$下，我们得到了$n\geq3$中$a\in (-1,0)$正轴对称解的一些定性性质.特别地，我们建立了正轴对称解的渐近展开。

英文摘要：

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 $\frac{\partial u}{\partial ν^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. 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.

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