  We solve an energy minimization problem for local fields. As an application
of these results, we improve on lower bounds set by Bombieri and Zannier for
the limit infimum of the Weil height in fields of totally p-adic numbers and
generalizations thereof. In the case of fields with mixed archimedean and
non-archimedean splitting conditions, we are able to combine our bounds with
similar bounds at the archimedean places for totally real fields.
