Liouville Type Theorem For A Nonlinear Neumann Problem

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