  Consider the following nonlinear Neumann problem \[ \begin{cases}
\text{div}\left(y^{a}\nabla u(x,y)\right)=0, & \text{for
}(x,y)\in\mathbb{R}_{+}^{n+1}\\ \lim_{y\rightarrow0+}y^{a}\frac{\partial
u}{\partial y}=-f(u), & \text{on }\partial\mathbb{R}_{+}^{n+1},\\ u\ge0 &
\text{in }\mathbb{R}_{+}^{n+1}, \end{cases} \]
  $a\in(-1,1)$. A Liouville type theorem and its applications are given under
suitable conditions on $f$. Our tool is the famous moving plane method.
