{
  "abstract": "Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \\begin{equation*} u''+a(t)u^γ=0 \\end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $γ\\in\\mathbb{R}\\setminus\\left\\{-1,0,1\\right\\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $γ=-1$. More generally, for every $p>1$, we consider the equation \\begin{equation*} \\left(\\left\\lvert u'\\right\\rvert^{p-2}u'\\right)'+a(t)u^γ=0 \\end{equation*} under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all $γ\\in\\mathbb{R}\\setminus\\left\\{0,p-1\\right\\}$. The necessary mean-value condition $γ\\int_{0}^{T}a(t)\\,\\mathrm{d} t<0$ is also sufficient for existence when $γ\\leq-1$ or $γ>p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $γ\\in(-1,p-1)\\setminus\\{0\\}$. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06068",
  "author_total": 1,
  "authors": [
    {
      "name": "Alberto Cagnetta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8msd7sybzgqfx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06068",
  "fulltext_url": null,
  "id": "8msd7sybzgqfx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06068",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Nonlinear Differential Equations Analysis"
    ]
  ],
  "openalex": "W7220673567",
  "pdf_url": "https://arxiv.org/pdf/2610.06068",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8msd7sybzgqfx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06068",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06068"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A uniqueness result for two-step indefinite $p$-Laplacian equations",
  "title_text": "A uniqueness result for two-step indefinite p-Laplacian equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/8msd7sybzgqfx",
  "venue": "arXiv",
  "versions": []
}