{
  "abstract": "We introduce the nonlocal zero-mass equation \\[ - Δu + Φ_u(|x|)\\, |u|^{p-2}\\, u = f(u) \\quad \\text{in } {\\mathbb R}^N, \\qquad Φ_u(r) = a \\int_r^\\infty ρ^{-b}\\, h_u^{q-1}(ρ)\\, dρ, \\] where $h_u(ρ)$ is the mass of $|u|^p$ in the ball of radius $ρ$. For radial functions, this equation includes the defocusing inverse-power Schrödinger equation, the Chern-Simons-Schrödinger equation, and the Schrödinger-Poisson-Slater equation as special cases. We develop the associated ball-mass Lebesgue and Sobolev spaces, which are uniformly convex, and prove sharp compact radial embeddings above a new critical exponent, a Brézis-Lieb type splitting, and a Pohožaev identity. Exploiting a scaling invariance of the operator, we construct an unbounded sequence of eigenvalues of a scaled eigenvalue problem using the Fadell-Rabinowitz cohomological index, and obtain existence and multiplicity results in the subscaled, superscaled, and critical regimes. Our main result is a Brézis-Nirenberg type theorem, proved by means of a new scaled linking theorem, which gives a nontrivial radial solution for every $λ> 0$ that is not an eigenvalue. Specializing to the three models recovers several known results in a unified way and yields new ones, including a Brézis-Nirenberg type result for the Schrödinger-Poisson-Slater equation in dimensions $N \\ge 4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26585",
  "author_total": 1,
  "authors": [
    {
      "name": "Kanishka Perera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8mbrb5wq43rj5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26585",
  "fulltext_url": null,
  "id": "8mbrb5wq43rj5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26585",
  "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"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214102136",
  "pdf_url": "https://arxiv.org/pdf/2609.26585",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8mbrb5wq43rj5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26585",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26585"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A unifying zero-mass equation",
  "title_text": "A unifying zero-mass equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/8mbrb5wq43rj5",
  "venue": "arXiv",
  "versions": []
}