Harmonic singularity at infinity in \(\mathbb{R}^n\) (Q1978883)

The main result of this paper is the following theorem: Let \(u\) be a harmonic function defined outside a compact set in \(\mathbb{R}^n\), \(n\geq 2\). Then the following are equivalent: 1) \(u(x)=o(|x|)\) when \(|x|\to \infty \). 2) The limes inferior of \(u(x)/|x|\) for \(|x|\to \infty \) is nonnegative. 3) There exists a locally integrable function \(\varphi \) such that \(u\geq \varphi \) outside a compact set, and the mean-value of \(\varphi (x)\) on \(|x|=r\) is \(o(r)\) when \(r\to \infty \). 4) There is a finite limit of \(u(x)\) for \(|x|\to \infty \) if \(n\geq 3\) and there is a finite limit of \((u(x)-\alpha \log |x|)\) for \(|x|\to \infty \) and some \(\alpha \) if \(n=2\) . From this theorem is deduced a generalized form of Liouville theorem in \(\mathbb{R}^n\) which is known to be equivalent to an improved version of the classical Bôcher theorem on harmonic point singularities.