Year: 2026
Type: preprint
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 $-Δ_g u=u^p$, $10$ 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 $10$, 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.
Source: arXiv (Cornell University)
Authors: Liang, Xiuda, Wang, Wenjie
Institution: Central China Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221790481