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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 $ψ(x^0,x) = e^{-iΩx^0}χ(x)$ of the nonlinear Dirac equation {equation*} i\underset{μ=0}{\overset{3}{\sum}} γ^μ\partial_μψ- mψ- F(\barψψ)ψ= 0 {equation*} where $Ω>m>0,$ $χ$ 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 $Ω.$ 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.