Game theoretical asymptotic mean value properties for non-homogeneous $p$-Laplace problems

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 $\Delta_p u:=\nabla\cdot(|\nabla u|^{p-2} \nabla u) = f$ 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 $\Delta_p^{\text{N}}u:=|\nabla u|^{2-p} \Delta_p u = 0$, which yields the 1-homogeneous normalized $p$-Laplacian, such normalization is not applicable when $f \neq 0$. 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.