{
  "abstract": "Let $w_0$ be the positive homoclinic solution of $-w''+w=w^p$ and set $c_p=w_0(0)$. We study bounded positive solutions of \\[ -Δv+v=v^p\\quad\\text{in }\\mathbb R^N_+, \\qquad v=c_p\\quad\\text{on }\\partial\\mathbb R^N_+, \\qquad v(x',x_N)\\to0 \\] uniformly as $x_N\\to\\infty$. This is the threshold boundary value left open by Fernández and Weth [Math.\\ Ann.\\ \\textbf{383} (2022), 361--397]. We prove a dimension-dependent rigidity/nonuniqueness picture. For every $p>1$, the profile $w_0(x_N)$ is the unique bounded positive solution when $2\\le N\\le5$, whereas for $N\\ge8$ there is a one-parameter family of bounded positive non-one-dimensional solutions. In the critical dimension $N=7$, nonuniqueness holds whenever a cubic coefficient $κ(p)$ of the exact reduced nonlinearity is positive; in particular $κ(p)>0$ for all $p\\ge2$, and continuity extends this range below $2$. The corresponding amplitudes have Fowler ends, with their limiting energy, neck size, bulge size, and logarithmic period determined to leading order. The case $N=6$, as well as the complementary range in $N=7$, remain open. The proof combines a boundary-adapted Modica estimate, projection onto the kernel of the one-dimensional linearized operator, an automatic tail non-concentration estimate, and a nonlinear cell problem yielding the exact reduced nonlinearity. For $N\\ge8$ the reduced linearization is invertible in weighted spaces. In $N=7$ the critical resonance is bypassed by solving the projected radial equation nonlinearly and closing the transverse equation by a Schauder--Tychonoff fixed point.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02790",
  "author_total": 1,
  "authors": [
    {
      "name": "Phuong Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2kmfns2bw4c2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02790",
  "fulltext_url": null,
  "id": "2kmfns2bw4c2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02790",
  "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": "W7220425322",
  "pdf_url": "https://arxiv.org/pdf/2610.02790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2kmfns2bw4c2p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition",
  "title_text": "The critical boundary value for a nonlinear Schrödinger equation in a half-space: rigidity and dimensional transition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/2kmfns2bw4c2p",
  "venue": "arXiv",
  "versions": []
}