A Strong Maximum Principle and a Compact Support Principle for infinity Laplacian

In this article we find necessary and sufficient conditions for the strong maximum principle and compact support principle for non-negative solutions to the quasilinear elliptic inequalities $${\mathrm{\Delta }}_{\mathrm{\infty }}u+G(\mathrm{\mid }Du\mathrm{\mid })-f(u)\le 0\text{in}\mathcal{O},$$ and $${\mathrm{\Delta }}_{\mathrm{\infty }}u+G(\mathrm{\mid }Du\mathrm{\mid })-f(u)\ge 0\text{in}\mathcal{O},$$ where $\mathcal{O}$ denotes the infinity Laplacian, $G$ is an appropriate continuous function and $f$ is a nondecreasing, continuous function with $f(0)=0$.