Strongly localized semiclassical states for nonlinear Dirac equations

Thomas Bartsch; Tian Xu

doi:10.3934/dcds.2020297

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2021, Volume 41, Issue 1: 29-60. Doi: 10.3934/dcds.2020297

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Strongly localized semiclassical states for nonlinear Dirac equations

Thomas Bartsch^1, , and
Tian Xu^2,

1.
Mathematisches Institut, Universität Giessen, 35392, Giessen, Germany
2.
Center for Applied Mathematics, Tianjin University, 300072, Tianjin, China

^* Corresponding author: Thomas Bartsch
^* Corresponding author: Thomas Bartsch

Received: January 2020

Revised: May 2020

Early access: August 2020

Published: January 2021

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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Abstract

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.

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

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