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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
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