Archive for Rational Mechanics and Analysis

, Volume 216, Issue 2, pp 415–447 | Cite as

Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations

  • Yanheng Ding
  • Tian Xu


The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling:
$$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g (|w|) w.\end{array}$$
Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains \({\Lambda_j \subset \mathbb{R}^3}\) such that \({-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}\) , \({x_j\in\Lambda_j}\) , then the k-families of solutions \({w_\hbar^j}\) concentrate around x j as \({\hbar\to 0}\) , respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions.


Dirac Equation Dirac Operator Localize Concentration Ground State Solution Dirac System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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