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⋅(iħ∇+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∈R3x∈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−2uW(x)|u|p−2u, p∈(2,3)p∈(2,3), or critical: W1(x)|u|p−2u+W2(x)|u|uW1(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.