Strongly localized semiclassical states for nonlinear Dirac equations
Year: 2020
Type: article
Abstract: We study semiclassical states of the nonlinear Dirac equation \begin{document}$ -i\hbar{\partial}_t\psi = ic\hbar\sum\limits_{k = 1}^3{\alpha}_k{\partial}_k\psi - mc^2{\beta} \psi - M(x)\psi + f(|\psi|)\psi, \quad t\in \mathbb{R}, \ x\in \mathbb{R}^3, $\end{document} where \begin{document}$ V $\end{document} is a bounded continuous potential function and the nonlinear term \begin{document}$ f(|\psi|)\psi $\end{document} is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity \begin{document}$ f(s) = s^p $\end{document} . We develop a variational method for the strongly indefinite functional associated to the problem.
DOI: 10.3934/dcds.2020297
Authors: Thomas Bartsch, Tian Xu
Institution: Tianjin University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W3035481881