{
  "abstract": "We study $-Δ_p u = |u|^{q-1}u$ in $\\mathbb{R}^N$ with $N>p\\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^*-1<q<q_c(N,p)$ only for radial solutions, stating that without radial symmetry they were not able to conclude. The obstruction is that the arguments available for $p=2$ rest on a monotonicity formula with no monotone analogue when $p\\ne2$. We remove the radial hypothesis: every $C^1$ weak solution stable outside a compact set is trivial when $p^*-1<q<q_c(N,p)$. The whole range $p-1<q<q_c(N,p)$ with $q\\ne p^*-1$ is thereby settled, as it is for $p=2$ by Farina's theorem, and every finite Morse index solution in it is trivial. In its place we iterate the comparison principle against an explicit two-parameter supersolution, upgrading the decay forced by stability to that of the $p$-harmonic fundamental solution; this step uses no stability, only smallness of the self-similar decay, and is stated on its own. The upper threshold is sharp: for every $q>q_c(N,p)$ we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every $C^1$ solution stable outside a compact set has finite energy; for $p>2$ this removes the a priori $\\mathcal D^{1,p}$ assumption from the low-Morse-index classification of Farina, Mercuri and Willem and answers their question on the Morse index of the Aubin--Talenti extremals: it is one. At the upper endpoint, $q_c(N,p)$ is the exact threshold for a nonzero homogeneous weak solution on $\\mathbb R^N\\setminus\\{0\\}$ to be stable outside a ball, and at the threshold there are exactly two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02751",
  "author_total": 1,
  "authors": [
    {
      "name": "Phuong Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbzptf4pwr8zp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02751",
  "fulltext_url": null,
  "id": "bbzptf4pwr8zp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02751",
  "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": "W7220445155",
  "pdf_url": "https://arxiv.org/pdf/2610.02751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbzptf4pwr8zp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation",
  "title_text": "Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the p-Laplace Lane--Emden equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/bbzptf4pwr8zp",
  "venue": "arXiv",
  "versions": []
}