Strongly localized semiclassical states for nonlinear Dirac equations

Abstract

<p style='text-indent:20px;'>We study semiclassical states of the nonlinear Dirac equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a bounded continuous potential function and the nonlinear term <inline-formula><tex-math id="M2">\begin{document}$ f(|\psi|)\psi $\end{document}</tex-math></inline-formula> is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity <inline-formula><tex-math id="M3">\begin{document}$ f(s) = s^p $\end{document}</tex-math></inline-formula>. We develop a variational method for the strongly indefinite functional associated to the problem.