Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon Nonlinearities
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2025
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| author | Khuu, Tan-Dat Huynh, Trung-Hieu Vo, Hoang-Hung |
| author_facet | Khuu, Tan-Dat Huynh, Trung-Hieu Vo, Hoang-Hung |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon Nonlinearities Khuu, Tan-Dat Huynh, Trung-Hieu Vo, Hoang-Hung Analysis of PDEs 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. |
| title | Liouville--Type Results for Infinity Elliptic Equations Involving Gradient and Hardy--Hénon Nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.16116 |