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Nonrelativistic limit and some properties of solutions for nonlinear Dirac equations

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Abstract

In this paper, we study the nonrelativistic limit and some properties of solutions for the following nonlinear Dirac equation

$$\begin{aligned} -ic\sum _{k=1}^3 \alpha _k \partial _k \psi + m c^2 \beta \psi - \omega \psi =g(|\psi |)\psi , ~~\text {in}\ \mathbb {R}^3 \end{aligned}$$

where c denotes the speed of light, \(m>0\) is the mass of the electron. We show that solutions of nonlinear Dirac equation converge to the corresponding solutions of a coupled system of nonlinear Schrödinger equations as the speed of light tends to infinity for electrons with small mass. Moreover, we also prove the uniform boundedness and the exponential decay properties of the solutions for the nonlinear Dirac equation with respect to the speed of light c.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yanheng Ding, Xiaojing Dong & Qi Guo

  2. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Yanheng Ding, Xiaojing Dong & Qi Guo

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  1. Yanheng Ding
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  2. Xiaojing Dong
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  3. Qi Guo
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Correspondence to Qi Guo.

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Communicated by Andrea Malchiodi.

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Ding, Y., Dong, X. & Guo, Q. Nonrelativistic limit and some properties of solutions for nonlinear Dirac equations. Calc. Var. 60, 144 (2021). https://doi.org/10.1007/s00526-021-02038-x

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