Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains
Abstract
We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains Ωsubsetneq𝕊n, n≥2. Set pn*=2n/(2n−3). For the Lane--Emden equation −Δg u=up, 1<p≤ pn*, 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,pn*]. Every pair on this component satisfies ∇g2(u−(p−1)/3)>0 in Ω. For the torsion equation −Δg u=1, we establish ∇g2√u<0 for its unique Dirichlet solution in all dimensions n≥2, 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)=sp/(1+λsδ), with 1<p≤ pn*, λ>0, and 0<δ≤(p−1)/3, we also prove the existence of at least one positive solution u for which the transform (∫u∞ ds/f(s))1/3 has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.
Related
- Power convexity of the torsion function on horo-convex domains in hyperbolic space
- A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
- Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains
- Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains
- Concavity of the Ornstein--Uhlenbeck torsion function on centered ellipses and its failure on convex domains
BibTeX
@misc{liang2026transformed,
title = {{Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains}},
author = {Xiuda Liang and Wenjie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17012},
eprint = {2610.17012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17012},
}
RIS
TY - UNPB
TI - Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains
AU - Xiuda Liang
AU - Wenjie Wang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17012
UR - https://arxiv.org/abs/2610.17012
AB - We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains Ωsubsetneq𝕊^n, n≥2. Set p_n^*=2n/(2n−3). For the Lane--Emden equation −Δ_g u=u^p, 1<p≤ 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 ∇_g^2(u^−(p−1)/3)>0 in Ω. For the torsion equation −Δ_g u=1, we establish ∇_g^2√u<0 for its unique Dirichlet solution in all dimensions n≥2, 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≤ p_n^*, λ>0, and 0<δ≤(p−1)/3, we also prove the existence of at least one positive solution u for which the transform (∫_u^∞ ds/f(s))^1/3 has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.
ER -