  We numerically construct asymptotically $AdS_4$ solutions to
Einstein-Maxwell-dilaton theory. These have a dipolar electrostatic potential
turned on at the conformal boundary $S^2\times \mathbb{R}_t$. We find two
classes of geometries: $AdS$ soliton solutions that encode the full
backreaction of the electric field on the $AdS$ geometry without a horizon, and
neutral black holes that are "polarised" by the dipolar potential. For a
certain range of the electric field $\mathcal{E}$, we find two distinct
branches of the $AdS$ soliton that exist for the same value of $\mathcal{E}$.
For the black hole, we find either two or four branches depending on the value
of the electric field and horizon temperature. These branches meet at critical
values of the electric field and impose a maximum value of $\mathcal{E}$ that
should be reflected in the dual field theory. For both the soliton and black
hole geometries, we study boundary data such as the stress tensor. For the
black hole, we also consider horizon observables such as the entropy. At finite
temperature, we consider the Gibbs free energy for both phases and determine
the phase transition between them. We find that the $AdS$ soliton dominates at
low temperature for an electric field up to the maximum value. Using the
gauge/gravity duality, we propose that these solutions are dual to deformed
ABJM theory and compute the corresponding weak coupling phase diagram.
