Liouville theorems on the upper half space

In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a\in (0,1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a\mathrm{\nabla }u)=0$ on the upper half space.