{
  "abstract": "The following equation is considered: $$-Δ_p u(\\boldsymbol{x}) + Δ_q u(\\boldsymbol{x}) = f\\big(\\boldsymbol{x}, u(\\boldsymbol{x})\\big)\\text{ for } \\boldsymbol{x} \\in Ω, \\qquad u(\\boldsymbol{x}) = 0 \\text{ for } \\boldsymbol{x}\\in \\partial Ω.$$ We start by showing that the energy functional associated with the considered problem no longer has the properties typical of its non-competing counterpart, which prevents a direct application of standard variational and monotonicity methods. Using Ekeland's variational principle combined with properties imposed on the nonlinear term~$f$, we prove the existence of a nonzero generalized solution to the problem. It is also shown that in the one-dimensional case $Ω= (-1,1)$, assuming that $f$ satisfies the symmetry conditions $$f(x,u) = f(-x,u) = -f(x,-u),$$ together with the sign condition $f(x,u) \\geq 0$ for $u \\geq 0$ and an appropriate growth bound on $f$, a convex relaxation of the action functional yields the existence of a positive weak solution. Furthermore, we prove that in this setting, and under a slightly stronger sign condition $$f(x,u) > 0\\text{ for }u > 0,$$ all global minimizers of the action functional do not vanish in $(-1,1)$ but fail to be of class $C^1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08624",
  "author_total": 1,
  "authors": [
    {
      "name": "Michał Bełdziński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4vepwkx64xy5f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08624",
  "fulltext_url": null,
  "id": "4vepwkx64xy5f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08624",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08624",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4vepwkx64xy5f.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08624",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08624"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian",
  "title_text": "Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/4vepwkx64xy5f",
  "venue": "arXiv",
  "versions": []
}