# Liouville theorems on the upper half space

## TL;DR

This paper proves Liouville theorems for bounded solutions to specific linear elliptic equations on the upper half space, showing that constants are the only positive solutions under certain conditions.

## Contribution

It establishes new Liouville theorems for solutions to weighted elliptic equations on the upper half space, characterizing positive solutions as constants.

## Key findings

- Constants are the only positive solutions for the weighted elliptic equation with parameter a in (0,1).
- Solutions bounded from below are characterized as constants.
- Theorems extend understanding of elliptic equations on half spaces.

## 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.