Liouville Type Theorem For A Nonlinear Neumann Problem

Abstract: 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.