  Our purpose in this paper is to provide a self contained account of the
inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a
prescribed continuous data on the boundary of bounded domains. We employ a
combination of Perron's method and a priori estimates to give general
sufficient conditions on the right hand side $f$ that would ensure existence of
viscosity solutions to the Dirichlet problem. Examples show that these
sufficient conditions may not be relaxed. We also identify a class of
inhomogeneous terms for which the corresponding Dirichlet problem has no
solution in any domain with large in-radius. Several results, which are of
independent interest, are developed to build towards the main results. The
existence theorems provide substantial improvement of previous results,
including our earlier results \cite{BMO} on this topic.
