  Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform
exterior cone condition. We establish existence and uniqueness of continuous
solutions of the Dirichlet Problem associated to certain intrinsic nonlinear
mean value properties in $\Omega $. Furthermore we show that, when properly
normalized, such functions converge to the $p$-harmonic solution of the
Dirichlet problem in $\Omega$, for $p\in[2,\infty)$. The proof of existence is
constructive and the methods are entirely analytic, a fundamental tool being
the construction of explicit, $p$-independent barrier functions in $\Omega$.
