Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--H\'enon Nonlinearities

In this paper we study Liouville-type properties for a class of degenerate elliptic equations driven by the fractional infinity Laplacian with nonlinear lower-order terms, $$\Delta_\infty^{\beta}u - c\,H(u,\nabla u) - \lambda\, f(|x|,u)=0 \qquad \text{in }\mathbb{R}^n,$$ where $\beta\in[0,2]$, $\Delta_\infty^\beta$ denotes the fractional infinity Laplace operator, and the nonlinearities $H$ and $f$ represent Hamiltonian and Hardy--H\'enon type effects, respectively. We extend the Liouville theory for the classical and normalized infinity Laplacian by establishing a new weighted comparison principle together with sharp local Lipschitz estimates for viscosity solutions. Our Liouville theorems are derived from precise growth conditions for bounded nonnegative solutions when $f$ exhibits power-type behavior, i.e.\ $f\sim u^\gamma$. We also treat the exponential case $f\sim e^u$, for which the equation becomes strongly supercritical: under suitable assumptions on the growth of $u$ at spatial infinity, only partial Liouville-type conclusions can be obtained. The analysis relies on radial reduction, barrier constructions, and refined comparison arguments. Altogether, the results provide a unified framework linking regularity, comparison principles, and Liouville-type phenomena for degenerate elliptic equations involving fractional infinity Laplacians and nonlinear lower-order effects.