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 + N u = 0 on  ∂ R + N in low dimension, under suitable assumptions on A and g . Our method is based upon a combination of Fourier series and Liouville theorems.