## Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system.(English)Zbl 1326.35305

Summary: This paper is concerned with the following Maxwell-Dirac system $\begin{cases} - i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + M(x) u - K(x) \phi u = F_u(x, u), \\ - \Delta \phi = 4 \pi K(x) \mid u \mid^2, \end{cases} \text{ in } \mathbb{R}^3$ where $$M(x)$$ is a external potential and $$F(x, u)$$ is an asymptotically quadratic nonlinearity modeling various types of interactions. In view of the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, existence and multiplicity of stationary solutions are obtained for system without any periodicity assumption via variational methods.

### MSC:

 35Q41 Time-dependent Schrödinger equations and Dirac equations 35Q60 PDEs in connection with optics and electromagnetic theory 81V10 Electromagnetic interaction; quantum electrodynamics 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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