The asymptotic $p$-Poisson equation as $p\to \mathrm{\infty }$ in Carnot-Carathéodory spaces

In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\mathrm{\infty }$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\mathrm{\infty }-$Laplacian and the Eikonal equation.