Mean Value Properties for the Weinstein Equation Using the Hyperbolic Metric

Abstract

In this paper we consider solutions of the Weinstein equation $$\begin{aligned} \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}+\frac{\ell }{ x_{n}^{2}}u=0, \end{aligned}$$ on some open subset \(\Omega \subset \mathbb R ^{n}\cap \{x_{n}>0\}\) subject to the conditions \(4\ell \le (k+1)^{2}\). If \(l=0\), the operator \(x_{n}^{2k/n-2}\left( \Delta u-\frac{k}{x_{n}}\frac{\partial u}{\partial x_{n}}\right) \) is the Laplace–Beltrami operator with respect to the Riemannian metric \(ds^{2}=x_{n}^{-2k/n-2}\left( \sum _{i=1}^{n}dx_{i} ^{2}\right) \). In case \(k=n-2\) the Riemannian metric is the hyperbolic distance of Poincare upper half space. The Weinstein equation is connected to the axially symmetric potentials. The solutions of of the Weinstein equation form a so-called Brelot harmonic space and therefore it is known they satisfy the mean value properties with respect to the harmonic measure. We present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. The key idea is to transform the solutions to the eigenfunctions of the Laplace–Beltrami operator in the Poincare upper half-space model.