On multiplicity of semi-classical solutions to a nonlinear Maxwell-Dirac system

We study multiplicity of semi-classical solutions of the nonlinear Maxwell-Dirac system:α(ih+q(x)A→(x))w-aβw-ωw-q(x)φ(x)w=f(x,|w|)w-δφ=q(x)|w|2-δAk=q(x)(αkw)w-k=1,2,3 for x∈R3, where A→ is the magnetic field, φ is the electron field, q describes the changing pointwise charge distribution, and f describes the self-interaction which is either subcritical: W(x)|u|p-2u, p∈(2, 3), or critical: W1(x)|u|p-2u+W2(x)|u|u. The number of solutions obtained is described by the ratio of maximum and behavior at infinity of the potentials.