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.


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