Liouville theorems on the upper half space

Abstract

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 $div(x_n^a\nabla u)=0$ on the upper half space.

Liouville Theorems for a linear equation with Dirichlet or Neuman boundary condition are obtained. Surprisedly, no boundary condition is needed for Corollary 1.3. The linear equation is closely related to our recent study of extension operators, and is closely related to Grushian type operators as well