{
  "abstract": "We study the Neumann boundary value problem $$ -\\operatorname{div}\\left(\\frac{\\nabla u}{\\sqrt{1-|\\nabla u|^2}}\\right) = λa(x)g(u) \\quad \\text{in } Ω, \\qquad \\frac{\\nabla u}{\\sqrt{1-|\\nabla u|^2}}\\cdot \\mathbf n = 0 \\quad \\text{on } \\partialΩ, $$ where $Ω\\subset \\mathbb R^N$ is a bounded convex domain, $a\\in L^\\infty(Ω)$ is an indefinite weight with $\\int_Ωa(x) \\, dx< 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $λ>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_λ^{(l)}$ with negative energy, and a mountain-pass critical point $u_λ^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour as $λ\\to+\\infty$ of both families of solutions. For the ground states we prove that, up to a subsequence, $u_λ^{(l)}\\to u_\\infty$ uniformly, where $u_\\infty$ satisfies $\\|\\nabla u_\\infty\\|_{L^\\infty(Ω)}=1$ and is characterized by being a maximizer of $\\int_Ωa(x) G(u) \\, dx$ and a minimizer of the kinetic energy among all such maximizers. In the model case $g(u)=|u|^{p-2}u$, $p>2$, we also show that the mountain-pass energy level satisfies the upper bound $c_λ=O(λ^{-2/(p-2)})$, and that $u_λ^{(s)}\\to0$ in $C(\\overlineΩ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09894",
  "author_total": 1,
  "authors": [
    {
      "name": "Ricardo Ziegele"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8e5qqzace8tyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09894",
  "fulltext_url": null,
  "id": "8e5qqzace8tyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09894",
  "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": "W7212173016",
  "pdf_url": "https://arxiv.org/pdf/2609.09894",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8e5qqzace8tyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.09894",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09894"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour",
  "title_text": "Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/8e5qqzace8tyr",
  "venue": "arXiv",
  "versions": []
}