Existence of standing waves for dirac fields with singular nonlinearities

Abstract

We prove the existence of stationary states for nonlinear Dirac equations of the form

$$i\sum\limits_{\mu = 0}^3 {\gamma ^\mu \partial _\mu \psi - M\psi + F\left( {\bar \psi \psi } \right)\psi = 0,} $$

((E))

whereM>0 andF is a singular self-interaction. In particular, in the model case whereF(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)=e ^iωtϕ(x), wherex ₀=t andx=(x ₁,x ₂,x ₃), such that ϕ has compact support. IF 0<α<1/3, then ϕ is of classC ¹. If 1/3<α<1, then ϕ is continuously differentiable, except on some sphere {|x|=R}, where |∇ϕ| is infinite.