Symmetry and uniqueness of nonnegative solutions of some problems in the halfspace

We derive some 1-D symmetry and uniqueness or non-existence results for nonnegative solutions of {−div(A(x)∇u)=u−g(x)in R+Nu=0on ∂R+N in low dimension, under suitable assumptions on AA and gg. Our method is based upon a combination of Fourier series and Liouville theorems.