A counterexample to the pointwise validity of an asymptotic mean value property for $p$-harmonic functions

We give a counterexample showing that an asymptotic mean value property for $p$-harmonic functions in ${\mathbb{R}}^d$ does not necessarily hold in the pointwise sense for $d\ge 3$ and $p>2$ near $2$. The proof uses a perturbation argument from $p=2$ and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of $p$ for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.