Liouville theorem for poly-harmonic functions on ${{\mathbb {R}}}^{n}_{+}$

Abstract

In this paper, we will prove a Liouville theorem for poly-harmonic functions on ${{\mathbb {R}}}^{n}_{+}$ with Navier boundary conditions, that is, the nonnegative poly-harmonic functions u satisfying $u(x)=o(|x|^{3})$ at $\infty$ must assume the form

$$\begin{aligned} u(x)=C x_{n} \end{aligned}$$

in $\overline{{{\mathbb {R}}}^{n}_{+}}$, where $n\ge 2$ and C is a nonnegative constant. The assumption $u(x)=o(|x|^{3})$ at $\infty$ is optimal for us to derive the super poly-harmonic properties of u.