Existence and multiplicity of stationary solutions for a class of Maxwell–Dirac system

Abstract

This paper is concerned with the following Maxwell–Dirac system {−i∑k=13αk∂ku+aβu+M(x)u−K(x)ϕu=Fu(x,u),−Δϕ=4πK(x)∣u∣2,in R3 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.