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Standing wave solutions of Maxwell–Dirac systems

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Abstract

This paper is concerned with the following Maxwell–Dirac system

$$\begin{aligned}&-i\sum ^3_{k=1}\alpha _{k}\partial _{k}u + (V(x)+a)\beta u + \omega u-K(x)\phi u =F_u(x,u),\\&\quad -\Delta \phi =4\pi K(x)|u|^2, \end{aligned}$$

in \(\mathbb {R}^3\), where V(x) is a potential function and F(x, u) is a nonlinear function modeling various types of interaction and K(t, x) is the varying pointwise charge distribution. Since the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, we prove the existence of infinitely many geometrically distinct solutions for superquadratic as asymptotically quadtratic nonlinearities via variational approach. Some recent results in the literature are generalized and significantly improved. Some examples are also given to illustrate our main theoretical results.

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Acknowledgements

The author would like to thank the referees for their careful reading, critical comments and helpful suggestions, which helped to improve the quality of the paper.

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  1. Department of Mathematics, Higher Institute of Science Computer and Mathematics of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Abderrazek Benhassine

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  1. Abderrazek Benhassine
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Correspondence to Abderrazek Benhassine.

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Communicated by P. Rabinowitz.

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Benhassine, A. Standing wave solutions of Maxwell–Dirac systems. Calc. Var. 60, 107 (2021). https://doi.org/10.1007/s00526-021-01935-5

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