Inhomogeneous Dirichlet problems involving the infinity-Laplacian

Abstract

Our purpose in this paper is to provide a self-contained account of the inhomogeneous Dirichlet problem ${\Delta_\infty} u=f(x,u)$ where $u$ represents 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 a substantial improvement of previous results, including our earlier results [7] on this topic.