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 $Ω\subsetneq\mathbb S^n$, $n\ge2$. 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 compone