{
  "abstract": "Let $1 \\leq q < p$ and let $λ_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $Ω$. We study the energy functional $$ E_λ(u)=\\frac{1}{p}\\left(\\int_Ω|\\nabla u|^p\\,dx -λ\\int_Ω|u|^p\\,dx\\right)-\\mathcal{F}(u), \\quad u\\in W_0^{1,p}(Ω), $$ where $\\mathcal{F}$ is positively $q$-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between $\\mathcal{F}$ and the $κ$-th power of the principal part of $E_{λ_1}$ near this eigenspace, we describe the behavior of $E_{λ_1}$ according to the relations $pκ<q$, $pκ=q$, or $pκ>q$. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small $q$-homogeneous perturbations of $\\mathcal{F}$ influence the geometry of $E_λ$. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of $λ_1$ and two critical points in a right neighborhood of $λ_1$, which indicates an $S$-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25768",
  "author_total": 1,
  "authors": [
    {
      "name": "Vladimir Bobkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6dbb238vjcqde.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25768",
  "fulltext_url": null,
  "id": "6dbb238vjcqde",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25768",
  "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": "W7214078382",
  "pdf_url": "https://arxiv.org/pdf/2609.25768",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6dbb238vjcqde.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25768",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25768"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the generalized Fredholm alternative for the $p$-Laplacian with resonant subhomogeneous terms",
  "title_text": "On the generalized Fredholm alternative for the p-Laplacian with resonant subhomogeneous terms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/6dbb238vjcqde",
  "venue": "arXiv",
  "versions": []
}