Infinity Laplace equation with non-trivial right-hand side

Abstract

We analyze the set of continuous viscosity solutions of the infinity Laplace equation $-Delta^N_{infty}w(x) = f(x)$, with generally sign-changing right-hand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron's construction by means of a strict comparison principle. These extremal solutions are proved to be absolutely extremal solutions.

Keywords

• Infinity Laplace equation
• inhomogeneous equation
• viscosity solutions
• least solution
• greatest solution
• strict comparison principle
• existence
• uniqueness
• local Lipschitz continuity