Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian

We consider the (viscosity) solution $u^{\epsilon }$ of the elliptic equation ${\epsilon }^2{\mathrm{\Delta }}_p^{G}u=u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, ${\mathrm{\Delta }}_p^G$ is the {\it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $\varepsilon\to 0^+$involvingthevaluesof$u^\varepsilon$, in the spirit of Varadhan’s work \cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in \cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in \cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.