Uniqueness and nonuniqueness of p-harmonic Green functions on weighted Rⁿ and metric spaces

We study uniqueness of p-harmonic Green functions in domains Ω in a complete metric space equipped with a doubling measure supporting a p-Poincar\'e inequality, with 1<p<∞. For bounded domains in unweighted Rⁿ, the uniqueness was shown for the p-Laplace operator Δₚ and all p by Kichenassamy--V\'eron (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 x₀ has positive p-capacity, the Green function is a particular multiple of the capacitary potential for capₚ(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 x₀ 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 $p=2$.