Let Ω be a bounded domain in
ℝ
n
{\mathbb{R}^{n}}
. Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for
p
⩾
2
{p\geqslant 2}
. Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω.