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