Solutions of nonlinear Dirac equations

We study the Dirac equation:−i∂tψ=icℏ∑k=13αk∂kψ−mc2βψ+∇ψG(x,ψ), and obtain existence and multiplicity results of stationary solutions for several classes of nonlinearities G:R3×C4→R modeling various types of interaction. A typical result states that if G(x,u)G(x,u) depends periodically on x and is even in u, the problem has infinitely many geometrically different localized solutions. The arguments are variational. The associated Lagrangian functional is strongly indefinite and the Palais–Smale condition does not hold. We apply some recently developed critical point theorems.