Uniqueness of positive solutions to some nonlinear Neumann problems

Using the moving plane method, we obtain a Liouville type theorem for nonnegative solutions of the Neumann problem{div(ya∇u(x,y))=0,x∈Rn,y>0,limy→0+⁡yauy(x,y)=−f(u(x,0)),x∈Rn, under general nonlinearity assumptions on the function f:R→R for any constant a∈(−1,1).