Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials | x − y | − s with d − 2 ⩽ s < d . For a given axis-supported external field, the support and the density of the corresponding extremal measure on S d is determined. The special case s = d − 2 yields interesting phenomena, which we investigate in detail. A weak ∗ asymptotic analysis is provided as s → ( d − 2 ) + .