Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon 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, $${\mathrm{\Delta }}_{\mathrm{\infty }}^{\beta }u-cH(u,\mathrm{\nabla }u)-\lambda f(\mathrm{\mid }x\mathrm{\mid },u)=0\text{in }{\mathbb{R}}^n,$$ where $\beta \in [0,2]$, ${\mathrm{\Delta }}_{\mathrm{\infty }}^{\beta }$ denotes the fractional infinity Laplace operator, and the nonlinearities $H$ and $f$ represent Hamiltonian and Hardy–Hénon 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.