Stationary periodic solutions to Nonlinear Dirac equations with non-coercive potentials

Motivated by the Soler model, we study a nonlinear Dirac equation with a non-coercive Soler-type nonlinearity. Stationary periodic solutions are obtained via variational methods. This amounts to studying the equation on a three-dimensional torus, where the spectrum of the physical Dirac operator on the torus needs to be clarified. To overcome the failure of coercivity of the potential we use a coercive perturbation. By carefully analyzing the critical levels and the linking structures, uniform estimates for the perturbed critical points are obtained when the frequency is sufficiently close to the mass. This allows us to pass to the zero-perturbation limit and obtain a nontrivial periodic solution.The argument also extends to suitable scalar external fields.