{
  "abstract": "In this paper, we investigate the analytic properties of the Hardy-type operator $-Δ+ μH_0$ on the $d$-dimensional integer lattice graph $\\mathbb{Z}^d$, where $μ\\geq -1$ 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 $\\mathbb{Z}^d$ 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 $$ -Δu + μH_0 u \\geq W u^p \\quad \\text{in } \\mathbb{Z}^d, $$ where $p > 0$, $d \\geq 3$, $μ\\geq -1$, and $H_0, W \\in C(\\mathbb{Z}^d)$ are strictly positive potentials satisfying the asymptotic behaviors $H_0(x) \\sim |x|^{-2}$ and $W(x) \\sim |x|^θ$ as $|x| \\to \\infty$, with $θ\\in \\mathbb{R}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13933",
  "author_total": 3,
  "authors": [
    {
      "name": "Huyuan Chen"
    },
    {
      "name": "Bobo Hua"
    },
    {
      "name": "Wendi Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9067baggy8m6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13933",
  "fulltext_url": null,
  "id": "d9067baggy8m6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13933",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13933",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9067baggy8m6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13933",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13933"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Poisson problem with Hardy operator on lattice graph $ \\mathbb{Z}^d$ and the application to semilinear equations",
  "title_text": "Poisson problem with Hardy operator on lattice graph $ \\mathbb{Z}^d$ and the application to semilinear equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/d9067baggy8m6",
  "venue": "arXiv",
  "versions": []
}