Poisson problem with Hardy operator on lattice graph $ \mathbb{Z}^d$ and the application to semilinear equations
Abstract
In this paper, we investigate the analytic properties of the Hardy-type operator −Δ+ μH0 on the d-dimensional integer lattice graph ℤd, where μ≥ −1 and H0 is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green kernel estimates on ℤd and obtain a necessary and sufficient weighted summability condition characterizing the solvability of the nonnegative Poisson problem associated with this operator. Building upon this classification, we provide a complete characterization of the nonexistence of positive solutions to the semilinear elliptic inequality −Δu + μH0 u ≥ W up in ℤd, where p > 0, d ≥ 3, μ≥ −1, and H0, W ∈ C(ℤd) are strictly positive potentials satisfying the asymptotic behaviors H0(x) ∼ |x|−2 and W(x) ∼ |x|θ as |x| → ∞, with θ∈ ℝ.
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BibTeX
@misc{chen2026poisson,
title = {{Poisson problem with Hardy operator on lattice graph $ \mathbbZ^d$ and the application to semilinear equations}},
author = {Huyuan Chen and Bobo Hua and Wendi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13933},
eprint = {2609.13933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13933},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Poisson problem with Hardy operator on lattice graph $ \mathbb{Z}^d$ and the application to semilinear equations
AU - Huyuan Chen
AU - Bobo Hua
AU - Wendi Xu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13933
UR - https://arxiv.org/abs/2609.13933
AB - In this paper, we investigate the analytic properties of the Hardy-type operator −Δ+ μH_0 on the d-dimensional integer lattice graph ℤ^d, where μ≥ −1 and H_0 is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green kernel estimates on ℤ^d and obtain a necessary and sufficient weighted summability condition characterizing the solvability of the nonnegative Poisson problem associated with this operator. Building upon this classification, we provide a complete characterization of the nonexistence of positive solutions to the semilinear elliptic inequality −Δu + μH_0 u ≥ W u^p in ℤ^d, where p > 0, d ≥ 3, μ≥ −1, and H_0, W ∈ C(ℤ^d) are strictly positive potentials satisfying the asymptotic behaviors H_0(x) ∼ |x|^−2 and W(x) ∼ |x|^θ as |x| → ∞, with θ∈ ℝ.
ER -