{
  "abstract": "We develop a constructive method for solving semilinear elliptic equations $Δu(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply combinatorial criteria to prove solvability on leafless infinite trees, integer lattices, the triangular and hexagonal lattices, and a Cayley graph of the discrete Heisenberg group. We further establish solvability for a broad class of Cayley graphs of semidirect products $G\\cong\\mathbb Z\\ltimes_θH$. In particular, $Δu=e^u$ has infinitely many solutions on $\\mathbb Z^2$, but none of finite energy. We also extend the method to the bi-Laplacian under two-step coupling conditions, to the $p$-Laplacian under a unique-neighbor condition, and to magnetic Laplacians.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07994",
  "author_total": 3,
  "authors": [
    {
      "name": "Bobo Hua"
    },
    {
      "name": "Yong Lin"
    },
    {
      "name": "Haohang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a930gwqdhr5y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07994",
  "fulltext_url": null,
  "id": "2a930gwqdhr5y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07994",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212054875",
  "pdf_url": "https://arxiv.org/pdf/2609.07994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a930gwqdhr5y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Solvability of Semilinear Elliptic Equations on Infinite Graphs",
  "title_text": "Solvability of Semilinear Elliptic Equations on Infinite Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2a930gwqdhr5y",
  "venue": "arXiv",
  "versions": []
}