A generalization of the Liouville theorem to polyharmonic functions (Q5932002)

Let \(m,n\) be integers, \(m\geq 1\), \(n\geq 2\), and let \(u\) be a polyharmonic function of order \(m\) on \(\mathbb{R}^n\) (that is, \(u\in C^{2m} (\mathbb{R}^n)\) and \(\Delta^mu=0\) on \(\mathbb{R}^n\), where \(\Delta^m\) is the \(m\)th iterated Laplacian operator). Let \(\sigma\) denote surface measure on the sphere \(S(r)= \{x\in \mathbb{R}^n:|x|= r\}\). It is shown that if \(q>2m-2\) and \(\lim \inf_{r\to +\infty}r^{-q-n+1} \int_{S(r)} u^+d\sigma <+\infty\), then \(u\) is a polynomial of degree at most \(q\). The same result is proved by different methods in a paper of the reviewer [Hiroshima Math. J. 31, 367-370 (2001)]. Several authors have proved weaker versions of the result, obtaining the same conclusion but with more restrictive hypotheses.