Some existence and uniqueness results for infinity Laplace equations on infinite graphs

We study the Dirichlet problem of the following discrete infinity Laplace equation on unbounded subgraphs \begin{equation*} \Delta_{\infty}u(x):=\inf_{y\sim x}u(y)+\sup_{y\sim x}u(y)-2u(x)=f(x). \end{equation*} For the homogeneous case ($f=0$), the existence and uniqueness of sublinear solutions are established. This result is applied to prove the existence and uniqueness of sublinear solutions for the homogeneous (normalized) infinity Laplace equations on unbounded Euclidean domains. Uniqueness is also shown for the case $f \geq 0$ on trees.