We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the p-Laplacian operator for \(p>2\) . Specifically, we characterize viscosity solutions to the p-Laplace equation \(\Delta _pu :=\nabla \cdot (|\nabla u|^{p-2} \nabla u) = f\) with a nontrivial right-hand side f, through novel asymptotic mean value formulas. While asymptotic mean value formulas for the homogeneous case ( \(f = 0\) ) have been previously established, leveraging the normalization \(\Delta _p^{N }u:=|\nabla u|^{2-p} \Delta _pu = 0\) , which yields the 1-homogeneous normalized p-Laplacian, such normalization is not applicable when \(f \ne 0\) . Furthermore, the mean value formulas introduced here motivate, for the first time in the literature, a game-theoretical approach for non-homogeneous p-Laplace equations. We also analyze the existence, uniqueness, and convergence of the game values, which are solutions to a dynamic programming principle derived from the mean value property.
