{
  "abstract": "In this paper, we consider \\[ \\begin{cases} u=K*(|x|^αa_0(x)v^p)&\\quad\\hbox{in }\\mathbb R^n,\\\\ v=K*(|x|^βb_0(x)u^q)&\\quad\\hbox{in }\\mathbb R^n, \\end{cases} \\] where $n\\ge2$, $0<ρ<n$, $p,q>0$, and $α,β>-ρ$. The radial kernel is nonincreasing and satisfies $K(r)\\asymp r^{ρ-n}$. The coefficients satisfy $0<c_0\\le a_0(x),b_0(x)\\le C_0$ and $|x|(|\\nabla\\log a_0(x)|+|\\nabla\\log b_0(x)|)\\le L$. We prove nonexistence if \\[ \\frac{n+α-\\varepsilon_a}{p+1} +\\frac{n+β-\\varepsilon_b}{q+1}>n-ρ+\\varepsilon_K, \\] where the defects bound the radial logarithmic derivatives. No symmetry, finite-energy condition, or prescribed decay is assumed. The proof combines a localized virial identity with a source-layer pressure estimate whose constants are uniform under centered dilations, including the low-order range $0<ρ\\le1$. We also identify the necessary scaling range for existence. For radial weighted Newton systems with unequal powers, we prove that the combined coefficient defect has a sharp, unattained infimum equal to the subcritical gap. A fixed bound on the coefficient ratios forces a strictly larger defect, whereas suitable coefficient--solution pairs approach the threshold with excess at most $C/\\log M$. In the scalar Newton case, the threshold is explicit and the approximating family converges, after normalization, to an explicit critical weighted profile. The sharpness results concern the Newton kernel; no corresponding construction is asserted for general fractional kernels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31167",
  "author_total": 2,
  "authors": [
    {
      "name": "Shengbing Deng"
    },
    {
      "name": "Bo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbfbpjnn93369.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31167",
  "fulltext_url": null,
  "id": "bbfbpjnn93369",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31167",
  "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": "W7223399401",
  "pdf_url": "https://arxiv.org/pdf/2610.31167",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbfbpjnn93369.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31167",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31167"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems",
  "title_text": "Liouville Theorems and Sharp Homogeneity Defects for Weighted Potential Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bbfbpjnn93369",
  "venue": "arXiv",
  "versions": []
}