Riesz energy problems in unbounded sets of $\mathbb{R}^{d}$

We investigate Riesz energy problems on unbounded conductors in $\mathbb{R}^d$ in the presence of general external fields $Q$. We provide new sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\mathbb{R}^{d}$, embedded in $\mathbb{R}^{d+1}$, when the external field is created by the potential of a simple discrete measure $\nu$ outside of $\mathbb{R}^{d}$. An extension of a classical theorem by de La Vallée-Poussin is established which may be of independent interest.
This is a joint work with P. Dragnev, R. Orive, and E. B. Saff.