Coulomb equilibrium in the external field of an attractive-repellent pair of charges

The aim of this paper is to provide a complete analysis of the Coulomb equilibrium problem in the euclidean space $\mathbb{R}^d$, $d\geq2$, associated to the kernel $1/|x|^{d-2}$, with a non-convex external field created by an attractive-repellent pair of charges placed in $\mathbb{R}^{d+1} \setminus \mathbb{R}^d$. We consider the admissible setting, where the equilibrium measure is compactly supported, as well as the limiting weakly admissible setting, with a weaker external field at infinity, where the existence of the equilibrium measure still holds but possibly with an unbounded support. The main tools for our analysis are the notions of signed equilibrium and balayage of measures. We note that for certain configurations of charges and distances to the conductor, the support of the equilibrium measure is a shell (multidimensional annulus).