{
  "abstract": "We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains $Ω\\subsetneq\\mathbb S^n$, $n\\ge2$. Set $p_n^*=2n/(2n-3)$. For the Lane--Emden equation $-Δ_g u=u^p$, $1<p\\le p_n^*$, we prove that the set of positive solution pairs $(p,u)$ has a connected component containing the near-linear solution curve and projecting onto $(1,p_n^*]$. Every pair on this component satisfies $\\nabla_g^2(u^{-(p-1)/3})>0$ in $Ω$. For the torsion equation $-Δ_g u=1$, we establish $\\nabla_g^2\\sqrt{u}<0$ for its unique Dirichlet solution in all dimensions $n\\ge2$, without an additional diameter or circumradius assumption. In both settings, the solutions specified above have a unique nondegenerate maximum and strictly geodesically convex positive superlevel sets. For the non-power reaction $f(s)=s^p/(1+λs^δ)$, with $1<p\\le p_n^*$, $λ>0$, and $0<δ\\le(p-1)/3$, we also prove the existence of at least one positive solution $u$ for which the transform $\\left(\\int_u^\\infty ds/f(s)\\right)^{1/3}$ has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17012",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiuda Liang"
    },
    {
      "name": "Wenjie Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4a0qy9118raq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17012",
  "fulltext_url": null,
  "id": "d4a0qy9118raq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17012",
  "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": "W7221790481",
  "pdf_url": "https://arxiv.org/pdf/2610.17012",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d4a0qy9118raq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17012",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17012"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains",
  "title_text": "Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/d4a0qy9118raq",
  "venue": "arXiv",
  "versions": []
}