{
  "abstract": "In this paper, we study the coupled singular weighted elliptic system $$\\left\\{ \\begin{aligned} -\\operatorname{div}(|x|^{-2a}\\nabla u)+λ_1 \\frac{u}{|x|^{2a}}&=βp\\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\\frac{|u|^{2^{\\sharp}-2}u}{|x|^{b2^{\\sharp}}},\\quad\\text{in}~\\mathbb{R}^N, -\\operatorname{div}(|x|^{-2a}\\nabla v)+λ_2\\frac{v}{|x|^{2a}}&=βq\\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\\frac{|v|^{2^\\sharp-2}v}{|x|^{b{2^\\sharp}}},\\quad\\text{in}~\\mathbb{R}^N, \\displaystyle\\int_{\\mathbb{R}^N}\\frac{|u|^2}{|x|^{2a}} dx=ρ_1^2,\\quad &\\int_{\\mathbb{R}^N}\\frac{|v|^2}{|x|^{2a}}dx=ρ_2^2. \\end{aligned} \\right. $$ where $N\\ge3$, $β\\in\\mathbb{R}$, $ρ_1,ρ_2>0$, $\\max\\{0,\\tfrac{N-4}{2}\\}\\le a<\\tfrac{N-2}{2}$, $a<b<a+1$, $2^{\\sharp}:=\\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{\\sharp}$. Here $2^\\sharp$ is the critical exponent of the Caffarelli--Kohn--Nirenberg inequality, and $λ_1,λ_2\\in\\mathbb{R}$ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for $β\\le0$. For $β>0$ we obtain positive normalized ground states, with positive Lagrange multipliers, in the mass subcritical, mass critical, and mass supercritical regimes, for explicit ranges of $β$. In the mass supercritical regime, a threshold $β_0\\ge0$ appears: a ground state exists for every $β>β_0$ and does not exist for $0<β<β_0$. We give conditions on $p,q$ under which $β_0=0$, and we prove that $β_0>0$ when $p,q\\ge2$. Finally, we describe the ground states as $β\\to0^+$ and as $β\\to+\\infty$: after a dilation, they converge to the ground states of the limit system without critical terms, or one component vanishes and the other concentrates on an extremal of the Caffarelli--Kohn--Nirenberg inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11494",
  "author_total": 2,
  "authors": [
    {
      "name": "Asmita Rai"
    },
    {
      "name": "Divya Goel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4qwx7zcarwfxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11494",
  "fulltext_url": null,
  "id": "4qwx7zcarwfxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11494",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212308582",
  "pdf_url": "https://arxiv.org/pdf/2609.11494",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4qwx7zcarwfxa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11494",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11494"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent",
  "title_text": "Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/4qwx7zcarwfxa",
  "venue": "arXiv",
  "versions": []
}