Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon Nonlinearities

Abstract

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, \[ Δ_\infty^βu - c\,H(u,\nabla u) - λ\, f(|x|,u)=0 \qquad \text{in }\mathbb{R}^n, \] where $β\in[0,2]$, $Δ_\infty^β$ 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.\ $f\sim u^γ$. 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.

Theorems & Definitions (26)

• Definition 1: Viscosity solution
• Lemma 2: $\mathcal{O}(n)$ invariance of $\Delta_\infty^\beta$
• proof
• Lemma 3
• proof
• Theorem 4: Comparison principle
• proof
• Theorem 5: Existence and Uniqueness
• proof
• Lemma 6: Stability