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Solutions of a Nonlinear Dirac Equation with External Fields

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Abstract

We study the stationary Dirac equation

$$-ic\hbar{\sum^3_{k=1}}\alpha_k\partial_k u+mc^2\beta u+M(x) u= R_u(x,u),$$

where M(x) is a matrix potential describing the external field, and R(x, u) stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption. For ħ fixed our discussion includes the Coulomb potential as a special case, and for the semiclassical situation (ħ → 0), we handle the scalar fields. We obtain existence and multiplicity results of stationary solutions via critical point theory.

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Authors and Affiliations

  1. Institute of Mathematics, AMSS, Chinese Academy of Sciences, Zhongguancun, 100080, Beijing, People’s Republic of China

    Yanheng Ding

  2. Dip. di Matematica, Università Milano, Via Saldini 50, 20133, Milano, Italy

    Bernhard Ruf

Authors
  1. Yanheng Ding
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  2. Bernhard Ruf
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Correspondence to Bernhard Ruf.

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Communicated by C.A. Stuart

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Ding, Y., Ruf, B. Solutions of a Nonlinear Dirac Equation with External Fields. Arch Rational Mech Anal 190, 57–82 (2008). https://doi.org/10.1007/s00205-008-0163-z

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