{
  "abstract": "We establish a Brézis-Oswald-type spectral principle for semilinear equations \\[ -Lu=f(x,u), \\] where $L$ is the generator of a positive $C_0$-semigroup on $L^p$ with compact positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let $a_0$ and $a_\\infty$ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let $λ_1(a)$ denote the generalized principal eigenvalue associated with the perturbed operator $-L-a$. We prove that \\[ λ_1(a_0)<0<λ_1(a_\\infty) \\] implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If $y\\mapsto f(x,y)/y$ is strictly decreasing and $a_0$ is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.10570",
  "author_total": 1,
  "authors": [
    {
      "name": "Tomasz Klimsiak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0dv6se2wh2y6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.10570",
  "fulltext_url": null,
  "id": "0dv6se2wh2y6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.10570",
  "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": "W7160959064",
  "pdf_url": "https://arxiv.org/pdf/2605.10570",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0dv6se2wh2y6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2605.10570",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.10570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent",
  "title_text": "Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/0dv6se2wh2y6b",
  "venue": "arXiv",
  "versions": []
}