{
  "abstract": "We consider the following class of singular elliptic problems \\[ \\begin{cases} -Δu = -\\dfrac{u}{|u|^{β+1}}χ_{\\{|u|>0\\}} +λ|u|^{p-1}u, & \\text{in }Ω,\\\\[1mm] u=0, & \\text{on }\\partialΩ, \\end{cases} \\] where $Ω\\subset\\mathbb R^N$ is a bounded smooth domain, $N\\geq3$, $0<β,p<1$, and $λ>0$. The energy functional is not of class $C^1$ in $H_0^1(Ω)$, and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if $λ$ exceeds a critical value $\\barλ$, which we identify and locate strictly between two Rayleigh-type parameters $0<λ_*<\\barλ<λ^*<+\\infty$. This gives a sharp existence threshold. We also prove that for every $λ$ in a neighborhood on the left of $λ^*$, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at $λ^*$. The second has a strictly positive energy. For $λ<λ^*$ sufficiently close to $λ^*$, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08389",
  "author_total": 4,
  "authors": [
    {
      "name": "J. C. de Albuquerque"
    },
    {
      "name": "M. L. M. Carvalho"
    },
    {
      "name": "Edcarlos D. Silva"
    },
    {
      "name": "M. L. Silva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/df0q6pm5kc6v4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08389",
  "fulltext_url": null,
  "id": "df0q6pm5kc6v4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08389",
  "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": "W7220973818",
  "pdf_url": "https://arxiv.org/pdf/2610.08389",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/df0q6pm5kc6v4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08389",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08389"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Extremal parameters, multiplicity and nonexistence for a singular elliptic problem",
  "title_text": "Extremal parameters, multiplicity and nonexistence for a singular elliptic problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/df0q6pm5kc6v4",
  "venue": "arXiv",
  "versions": []
}