  In this paper we study the existence of vortex-type solutions for a system of
self-dual equations deduced from the mass-deformed
Aharony--Bergman--Jafferis--Maldacena (ABJM) model. The governing equations,
derived by Mohammed, Murugan, and Nastse under suitable ansatz involving fuzzy
sphere matrices, have the new feature that they can support only
non-topological vortex solutions. After transforming the self-dual equations
into a nonlinear elliptic $2\times 2$ system we prove first an existence result
by means of a perturbation argument based on a new and appropriate scaling for
the solutions. Subsequently, we prove a more complete existence result by using
a dynamical analysis together with a blow-up argument. In this way we establish
that, any positive energy level is attained by a 1-parameter family of vortex
solutions which also correspond to (constraint) energy minimizers. In other
words, we register the exceptional fact in a BPS-setting that, neither a
"quantization" effect nor an energy gap is induced upon the system by the rigid
"critical" coupling of the self-dual regime.
