The Emden-Fowler equation on a spherical cap of Sn

Let Sn⊂Rn+1, n≥3, be the unit sphere, and let SΘ⊂Sn be a geodesic ball with geodesic radius Θ∈(0,π). We study the bifurcation diagram {(Θ,‖U‖∞)}⊂R2 of the radial solutions of the Emden-Fowler equation on SΘΔSnU+Up=0inSΘ,U=0on∂SΘ,U>0inSΘ,where p>1. Among other things, we prove the following: For each p>pS≔(n−2)∕(n+2), there exists Θ̲∈(0,π) such that the problem has a radial solution for Θ∈(Θ̲,π) and has no radial solution for Θ∈(0,Θ̲). Moreover, this solution is unique in the space of radial functions if Θ is close to π. If pS