In this paper, we investigate Riesz energy problems on %closed (possibly unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $|x|\to\infty$ assumed in several previous studies. We provide 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 $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $\nu$ outside of $\R^{d}$. Simple cases where $\nu$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vall\'ee-Poussin, are established. These results are of independent interest.
* Joint work with Ramon Orive - University La Laguna, Ed Saff - Vanderbilt University, and Franck Wielonsky - Universit\'e Aix-Marseille