
A class of degenerate elliptic equations with nonlinear boundary conditions
We consider positive solutions of the problem
\begin{equation}
(*)\qquad
\left\{
\begin{array}{l}\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n},
\\
\frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n},
\\
\end{array}
\right.
\end{equation}
where $a\in (1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and
$\frac{\partial u}{\partial \nu^a}:=\lim_{x_{n}\rightarrow
0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case
$b=0$, it is associated to fractional Laplacian equation $(\Delta)^{s}u=u^{q}
$ in entire space $\mathbb{R}^{n1}$.
We obtain the existence of positive axially symmetric solutions
to ($*$) for the case $a\in
(1,0)$ in
$n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a2},
\;\;q\geq\frac{na}{n+a2}$.
The nonexistence is obtained for the case $a\in (1,0)$, $b\geq
0$ and any $p,~q\gt 1$ in $n=2$ as well.
Keywords:existence, nonexistence, positive solutions, degenerate elliptic equation, nonlinear boundary conditions, symmetry, monotonicity Categories:35D30, 35J70, 35J25 