# American Institute of Mathematical Sciences

September  2020, 40(9): 5373-5381. doi: 10.3934/dcds.2020231

## Liouville theorems on the upper half space

 1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3 Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USA

Received  October 2019 Revised  March 2020 Published  June 2020

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.

Citation: Lei Wang, Meijun Zhu. Liouville theorems on the upper half space. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5373-5381. doi: 10.3934/dcds.2020231
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