In this paper, we investigate Liouville properties for degenerate elliptic equations involving the infinity Laplace operator with nonlinear lower-order terms, of the form \(\begin{aligned} \Delta _\infty ^\beta u - cH(u,\nabla u) - \lambda f(|x|,u) = 0 \text { in } \mathbb {R}^n, \end{aligned}\) where \(\beta \in [0,2], \Delta _\infty ^\beta \) denotes the inhomogeneous infinity Laplacian, and the nonlinear terms H and f are Hamiltonian and Hardy-Hénon type models, respectively. Our work extends existing Liouville frameworks for the classical and normalized infinity Laplacian by developing a new weighted comparison principle and a local Lipschitz estimate. To obtain a Liouville–type theorem, we establish growth conditions for bounded nonnegative viscosity solutions when f corresponds to increasing power-type nonlinearities, that is, when f grows as \(u^\gamma \) . Furthermore, we analyze the case of an exponential nonlinearity f, which grows as \(e^u\) , showing that this problem exhibits a strongly supercritical behavior; under suitable growth assumptions on u at infinity, only partial Liouville–type conclusions can be obtained. Our techniques combine radial analysis, barrier constructions, and refined comparison arguments, providing a unified framework connecting regularity, comparison principle, and Liouville properties.
