Existence of nodal solutions for Dirac equations with singular nonlinearities

TL;DR

This paper proves the existence of infinitely many nodal solutions for a nonlinear Dirac equation with singular nonlinearities, using a shooting method, and explores the connection to the M.I.T. bag model as the nonlinearity parameter tends to zero.

Contribution

It introduces a novel approach to find infinitely many solutions for Dirac equations with singular nonlinearities and links these solutions to the M.I.T. bag model.

Findings

Existence of infinitely many solutions proven using a shooting method.

Solutions exhibit specific behavior as the nonlinearity parameter approaches zero.

Established connection between the nonlinear Dirac equations and the M.I.T. bag model.

Abstract

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 {equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi- m\psi - F(\bar{\psi}\psi)\psi = 0 {equation*} where $\Omega>m>0,$ $\chi$ 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 $\Omega.$ 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.