  Energy minimizing BPS equations and solutions are obtained for a class of
models in Maxwell-scalar theory, where an abelian electric charge is immersed
in an effective dielectric of a real scalar field. The first order BPS
equations are developed using the straightforward on-shell method introduced by
Atmaja and Ramadhan. Employment of an auxiliary function of the scalar field
allows a scalar potential that displays a tachyonic instability. Consequently,
a nontopological scalar soliton is found to form around the charge. Examples
and solutions are provided for (1) a point charge or sphere in a flat Minkowski
background, and (2) an overcharged compact object in a Reissner-Nordstrom
background. The solutions presented here for the former (Minkowski) case
recover those that have been previously obtained, while the latter solutions
are new BPS solutions.

High Energy Physics - Theory;General Relativity and Quantum Cosmology