{
  "abstract": "We study normalized solutions of the nonlinear Dirac equation \\[ \\begin{cases} -iα\\cdot\\nabla u+mβu=μu+|x|^{-b}\\bigl(|u|^{q-2}-|u|^{p-2}\\bigr)u&\\text{in }\\mathbb{R}^3,\\\\ \\displaystyle\\int_{\\mathbb{R}^3}|u|^2\\,dx=a, \\end{cases} \\] where $m>0$, $0<b<1$, $2<q<3-b$, and $q<p<\\infty$. For each fixed $p>q$ and all sufficiently small $a>0$, we prove the existence of a normalized solution $(u_a,μ_a)$ with $u_a\\in H^1(\\mathbb{R}^3;\\mathbb C^4)\\cap L^\\infty(\\mathbb{R}^3;\\mathbb C^4)$ and $μ_a\\in(0,m)$. This covers the weighted Sobolev critical and supercritical ranges $p\\ge3-b$. Our method combines a mass-preserving spectral reduction with a convex truncation and uniform $L^\\infty$ estimates. For solutions arising from reduced minimization, we determine the energy asymptotics and the rate at which $μ_a$ approaches $m$ as $a\\to0^+$. After rescaling, the upper two-component spinors converge in $H^{\\frac12}(\\mathbb{R}^3;\\mathbb C^2)$ to the set of normalized ground states of a weighted nonlinear Schrödinger equation with exponent $q$, while the lower two-component spinors converge to zero in the same space. We also derive an asymptotic formula in $L^2(\\mathbb{R}^3;\\mathbb C^2)$ for the rescaled lower two-component spinor in terms of the gradient of the limiting Schrödinger ground state.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39310",
  "author_total": 2,
  "authors": [
    {
      "name": "Hao Wang"
    },
    {
      "name": "Xiaojun Chang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/der6dtj445tf7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39310",
  "fulltext_url": null,
  "id": "der6dtj445tf7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39310",
  "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/2610.39310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/der6dtj445tf7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Normalized solutions and small mass asymptotics for a nonlinear Dirac equation with a singular weight and competing powers",
  "title_text": "Normalized solutions and small mass asymptotics for a nonlinear Dirac equation with a singular weight and competing powers",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/der6dtj445tf7",
  "venue": "arXiv",
  "versions": []
}