{
  "abstract": "In this paper, we investigate nonexistence results for stable solutions of the semilinear elliptic problem $$(-Δ)^s u+λ|x|^αu=|x|^β|u|^{p-1}u \\; \\mbox{in}\\; \\mathcal{H}, $$ where $\\mathcal{H}=\\mathbb{R}^n$ or $\\mathcal{H}=\\mathbb{R}^n_+=\\{x=(x',x_n),\\, x'\\in\\mathbb{R}^{n-1},x_n>0\\}$ , with Dirichlet conditions in the half-space case. We assume that $n\\geq2s$, $p>1$, $λ>0,$ $α, β>-2s$ and $0<s\\leq1$. Our aim is to analyze the influence of the Hardy-type power potential $λ|x|^αu$ in the presence of the Hénon weight $|x|^β$, with particular emphasis on the existence and classification of weak stable solutions, including solutions that may be unbounded or sign-changing. We establish nonexistence results for nontrivial stable solutions and, more generally, for solutions that are stable outside a compact set, covering critical and supercritical regimes without additional assumptions, and the subcritical regime under a suitable $L ^\\infty$-smallness condition on a weighted version of the solution. Our results apply to both the local case $s = 1$ and the nonlocal case $0 < s < 1 $, in the whole space and in the half-space. The proofs are based on energy estimates combined with a Pohozaev identity. In the supercritical regime, we further establish a monotonicity formula, following the approach of Dávila et al. (Trans. Amer. Math. Soc. 369, 6087-6104, 2017).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05841",
  "author_total": 3,
  "authors": [
    {
      "name": "Habib Fourti"
    },
    {
      "name": "Abdelbaki Selmi"
    },
    {
      "name": "Cherif Zaidi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22ymw7r4zjavb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05841",
  "fulltext_url": null,
  "id": "22ymw7r4zjavb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05841",
  "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": "W7220715388",
  "pdf_url": "https://arxiv.org/pdf/2610.05841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22ymw7r4zjavb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases",
  "title_text": "The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/22ymw7r4zjavb",
  "venue": "arXiv",
  "versions": []
}