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Solutions of a nonlinear Dirac equation with external fields. (English) Zbl 1161.35041
Summary: We study the stationary Dirac equation $-ic\hbar \sum^3_{k=1} \alpha_k\partial_ku + mc^2\beta u + M(x)u = R_u(x,u),$ 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 $$\hbar$$ fixed our discussion includes the Coulomb potential as a special case, and for the semiclassical situation ($$\hbar \rightarrow 0$$), we handle the scalar fields. We obtain existence and multiplicity results of stationary solutions via critical point theory.

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35A15 Variational methods applied to PDEs
##### Keywords:
nonlinear Dirac equation; Coulomb potential
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##### References:
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