Existence of nodal solutions for Dirac equations with singular nonlinearities
Résumé
We prove, by a shooting method, the existence of infinitely many solutions of the form $\psi(x^0,x) = e^{-i\Omega x^0}\chi(x)$ of the nonlinear Dirac equation \begin{equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi- m\psi - F(\overline{\psi}\psi)\psi = 0 \end{equation*} where $\Omega>m>0,$ $\chi$ is compactly supported and \[ F(x) = \left\{\begin{array}{ll} p|x|^{p-1} & \text{if}~ |x|>0\\ 0 & \text{if}~ x=0 \end{array}\right. \] with $p\in(0,1),$ under some restrictions on the parameters $p$ and $\Omega.$ 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.
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