Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted $\mathbf{R}^n$ and metric spaces
Anders Björn, Jana Björn, Sylvester Eriksson-Bique, Xiaodan Zhou
TL;DR
The article addresses whether p-harmonic Green functions are unique in domains of complete metric spaces with doubling measures and p-Poincaré inequalities. It provides a new sufficient condition for uniqueness when the p-capacity of the singular point is zero, expressed via a growth limsup, and proves that the allowable range of p can be a nondegenerate interval. It also delivers a groundbreaking nonuniqueness example in a weighted Euclidean setting for p ≠ 2 and, separately, a nonuniqueness example for p = 2 in a nonstandard metric space, highlighting the delicate dependence on the ambient geometry and measure. The results bridge known uniqueness in Ahlfors regular and positive-capacity cases with new phenomena in general metric spaces, and they connect to Cheeger p-harmonic theory while clarifying when Green functions arise as capacitary potentials. These findings advance nonlinear potential theory on non-smooth spaces and illuminate the limits of Green function uniqueness beyond classical Euclidean settings.
Abstract
We study uniqueness of $p$-harmonic Green functions in domains $Ω$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $Δ_p$ and all $p$ by Kichenassamy--Véron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequence of the linearity of the Laplace operator $Δ$. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors $p$-regular spaces, as shown by Bonk--Capogna--Zhou (arXiv:2211.11974). When the singularity $x_0$ has positive $p$-capacity, the Green function is a particular multiple of the capacitary potential for $\text{cap}_p(\{x_0\},Ω)$ and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of $p$ for which it holds (while $x_0$ has zero $p$-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for $p=2$.