Existence of nodal solutions for Dirac equations with singular nonlinearities

We prove, by a shooting method, the existence of infinitely many solutions of the form $\psi (x^0,x)=e^{-i\mathrm{\Omega }{x}^0}\chi (x)$ of the nonlinear Dirac equation {equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi- m\psi - F(\bar{\psi}\psi)\psi = 0 {equation*} where $\mathrm{\Omega }>m>0,$ $\chi$ is compactly supported and [F(x) = {{array}{ll} p|x|^{p-1} & $\text{if}$ |x|>0 0 & $\text{if}$ x=0 {array}.] with $p\in (0,1),$ under some restrictions on the parameters $p$ and $\mathrm{\Omega }\mathrm{.}$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones.