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

• DOI: 10.1007/s00013-020-01464-1
• Publication Date: 2020-04-15T15:03:11Z
• Authors: Wei Dai; Guolin Qin
• Keywords: Harmonic; Constant (computer programming)

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.