{
  "abstract": "Let $N\\geq 3$, $2<q<p<2^*$, and let $V\\in L^\\infty(\\mathbb{R}^N)$ be real-valued and $\\mathbb{Z}^N$-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator $L=-Δ+V$. We prove that there exists $λ_\\infty>0$ such that, for every $λ\\geqλ_\\infty$, the competing-power equation $$ -Δu+V(x)u=|u|^{p-2}u-λ|u|^{q-2}u \\quad\\text{in }\\mathbb{R}^N $$ has a nontrivial solution $u_λ\\in H^1(\\mathbb{R}^N)$. Moreover, $$ \\|u_λ\\|_{H^1(\\mathbb{R}^N)}+\\|u_λ\\|_{L^\\infty(\\mathbb{R}^N)} \\leq Cλ^{-1/(q-2)}. $$ We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation $Lv=-|v|^{q-2}v$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23446",
  "author_total": 1,
  "authors": [
    {
      "name": "Bartosz Bieganowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vft4q08zc8xz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23446",
  "fulltext_url": null,
  "id": "9vft4q08zc8xz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23446",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7214003779",
  "pdf_url": "https://arxiv.org/pdf/2609.23446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vft4q08zc8xz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation",
  "title_text": "Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/9vft4q08zc8xz",
  "venue": "arXiv",
  "versions": []
}