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On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system

  • Yanheng Ding
  • Tian Xu
Article

Abstract

We study the semi-classical ground states of the nonlinear Maxwell–Dirac system:
$$\begin{aligned} \left\{ \begin{array}{l} \alpha \cdot \left( i\hbar \nabla + q(x)\mathbf{A }(x)\right) w-a\beta w -\omega w - q(x)\phi (x) w = P(x)g(\left| w\right| ) w\\ -\Delta \phi =q(x)\left| w\right| ^2\\ -\Delta {A_k}=q(x)(\alpha _k w)\cdot \bar{w}\ \ \ \ k=1,2,3 \end{array} \right. \end{aligned}$$
for \(x\in \mathbb{R }^3\), where \(\mathbf{A }\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hbar \) small. We also describe the concentration behavior of the solutions as \(\hbar \rightarrow 0\).

Mathematics Subject Classification (2000)

35Q40 49J35 

Notes

Acknowledgments

Special thanks to the reviewer for his/her good comments and suggestions. The comments were all valuable and helpful for revising and improving our paper, as well as to the important guiding significance of our research. The work was supported by the National Science Foundation of China (NSFC11331010, 10721061, 11171286).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Mathematics, AMSSChinese Academy of SciencesBeijingChina

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