A form of classical Liouville theorem for polyharmonic functions (Q1588310)

The authors prove the following version of Liouville's theorem for polyharmonic functions: If \(m\) is a positive integer and \(s\) is any real number with \(s>2m-2\), then \(u\), a polyharmonic function of order \(m\) is a polyharmonic polynomial of degree less than \(s\) if and only if there exists an increasing divergent sequence \(\left( r_i\right) _{i\geq 1}\) of positive numbers such that \[ \lim \inf _{i\rightarrow \infty }\left( \min _{\left|x\right|=r_i}\frac{ u\left( x\right) }{\left|x\right|^s}\right) \geq 0 . \]