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 $Ω\subsetneq\mathbb S^n$, $n\ge2$. 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.