Nonrelativistic limit and some properties of solutions for nonlinear Dirac equations

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 Schrodinger 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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