Existence of standing waves for Dirac fields with singular nonlinearities

We prove the existence of stationary states for nonlinear Dirac equations of the form i Σμ=03 γμdμψ-Mψ+F(anti ψψ)ψ=0, (E) where M>0 and F is a singular self-interaction. In particular, in the model case where F(s)=-s-α, for some 0<α<1, and for every ω>M, there exists a solution of (E) of the form ψ(t, x)=esup(iωt)φ(x), where x0=t and x=(x1, x2, x3), such that φ has compact support. If 0<α<1/3, then φ is of class C1. If 1/3<α<1, then φ is continuously differentiable, except on some sphere {vertical strokexvertical stroke=R}, where vertical stroke∇φvertical stroke is infinite. (orig.)