Necessity of B > n + 2 + 2ℓ for the spherical Onsager equation

Ascertain whether the inequality B > n + 2 + 2ℓ is necessary for the existence of solutions to the stereographically projected spherical Onsager mean field equation −Δu(x) = 4π B |x|^{n} K(x) e^{u(x)} in R^2 with normalization ∫_{R^2} |x|^{n} K(x) e^{u(x)} dx = 1, in the parameter regimes described after Corollary 5.7.

Background

After deriving Corollary 5.7, the authors use the Pohozaev identity to obtain constraints and show B must be less than n+2 under certain conditions. They point out that in some regimes it is unclear whether the stronger inequality B > n + 2 + 2ℓ is a necessary condition for existence.

Clarifying this necessity would sharpen the parameter thresholds for existence in the spherical Onsager vortex model, a mean field equation on the sphere transferred to R^2 via stereographic projection.