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We consider the nonlinear Dirac equation
The potential function V satisfies the conditions that the essential spectrum of the Dirac operator 
is 
and this Dirac operator has infinitely many eigenvalues in (−1, 1) accumulating at 1. This potential function V may change sign in ℝ3 and contains the classical Coulomb potential V (x) = −γ/|x| with γ > 0 as a special case. The nonlinearity F satisfies the resonance-type condition lim
. Under some additional conditions on V and F, we prove that this equation has infinitely many solutions.
By making use of Merle's general shooting method we investigate Dirac equations of the form
Here it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.
