An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations (Q1751596)

The authors address a stationary three-dimensional Dirac equation which contains an external-potential term with a local energy minimum. The equation is either a linear one, or with nonlinearity \(\sim |u|u\). In the semi-classical limit, which corresponds to a small coefficient in front of the first-derivative terms in the Dirac equation, they provide a rigorous proof of the existence of an infinite number of bound states in a vicinity of the energy minimum (in previous works, the proof was produced only for the existence of the ground state, or a of a finite number of bound states, in a similar setting). The analysis is related to that addressing bound states in the Schrödinger equation, also in the semi-classical limit, and makes use of a variational approach.