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Strongly localized semiclassical states for nonlinear Dirac equations

  • * Corresponding author: Thomas Bartsch

    * Corresponding author: Thomas Bartsch 
The second author is supported by the National Science Foundation of China (NSFC 11601370, 11771325) and the Alexander von Humboldt Foundation of Germany
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  • We study semiclassical states of the nonlinear Dirac equation

    $ -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, $

    where $ V $ is a bounded continuous potential function and the nonlinear term $ f(|\psi|)\psi $ 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 $ f(s) = s^p $. We develop a variational method for the strongly indefinite functional associated to the problem.

    Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.

    Citation:

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