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Solutions of nonlinear Dirac equations. (English) Zbl 1126.49003
Existence and multiplicity results for stationary solutions of the Dirac equation \(-i\partial _{t}\psi =ic\hslash \sum_{k=1}^{3}a_{k}\partial _{k}\psi -mc^{2}\beta \psi +\nabla _{\psi }G(x,\psi )\) are established via variational method. The associated Lagrangian functional is strongly indefinite and the Palais-Smale (PS) condition is not satisfied. Some recent saddle point theorems with Cerami type (PS) condition are applied to the considered functional.

MSC:
49J35 Existence of solutions for minimax problems
35Q40 PDEs in connection with quantum mechanics
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