The Dirac equation with interaction in the mass is
In this case the potential does not correspond to an electric potential, nevertheless, it is often used in the literature.
Something that may initially look surprising is the fact that this type of interaction supports bounded states for monotonically increasing potentials as shown in the figure below
This has been reported in the literature for sometime but without much of an explanation.
The deeper insight came from the study of the Dirac equation in solid state physics where this type of system was recognized as the result of the contact of two topologically distinct phases: one with positive mass and the other one with negative mass. This contact gives origin to the so called zero mode states that have many interesting features and are the subject of intensive research today.
I published a generalization of this bounded state with additional nonlinear interactions in my recent paper at PRL