Transformed Convexity for Lane--Emden Equations and Torsion on Spherical Domains

We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains $\mathrm{\Omega }⊊{\mathbb{S}}^n$, $n\ge 2$. Set $p_n^{\ast }=2n\mathrm{/}(2n-3)$. For the Lane–Emden equation $-{\mathrm{\Delta }}_{g}u=u^p$, $1<p\le p_n^{\ast }$, we prove that the set of positive solution pairs $(p,u)$ has a connected component containing the near-linear solution curve and projecting onto $(1,p_n^{\ast }]$. Every pair on this component satisfies ${\mathrm{\nabla }}_g^2(u^{-(p-1)\mathrm{/}3})>0$ in $\mathrm{\Omega }$. For the torsion equation $-{\mathrm{\Delta }}_{g}u=1$, we establish ${\mathrm{\nabla }}_g^2\sqrt{u}<0$ for its unique Dirichlet solution in all dimensions $n\ge 2$, 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 $f(s)=s^p\mathrm{/}(1+\lambda s^{\delta })$, with $1<p\le p_n^{\ast }$, $\lambda >0$, and $0<\delta \le (p-1)\mathrm{/}3$, we also prove the existence of at least one positive solution $u$ for which the transform ${({\int }_u^{\mathrm{\infty }}ds\mathrm{/}f(s))}^{1\mathrm{/}3}$ has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.