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Existence of Nodal Solutions for Dirac Equations with Singular Nonlinearities

Le Treust, Loïc (2013), Existence of Nodal Solutions for Dirac Equations with Singular Nonlinearities, Annales Henri Poincaré, 14, 5, p. 1383-1411. http://dx.doi.org/10.1007/s00023-012-0224-6

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Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00605824/fr/
Date
2013
Journal name
Annales Henri Poincaré
Volume
14
Number
5
Publisher
Springer
Pages
1383-1411
Publication identifier
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Author(s)
Le Treust, Loïc cc
Abstract (EN)
We prove by a shooting method the existence of infinitely many nodal solutions of the form $\psi(x^0,x) = e^{-i\Omega x^0}\chi(x)$ for nonlinear Dirac equations: \begin{equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi- m\psi - p|\overline{\psi}\psi|^{p-1}\psi = 0. \end{equation*} with $m>0$, $p\in(0,1)$ and $\chi(x)$ compactly supported under some restrictive conditions over $p$ and the frequency $\Omega>m$. We then study their behavior as $p$ tends to zero to establish the link between theses solutions and the M.I.T. bag model ones.
Subjects / Keywords
Exited states; M.I.T. bag model; Winding number; Shooting method; Nodal solutions