Infinitely many solutions of Dirac equations with concave and convex nonlinearities

We consider non-periodic Dirac equations with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. For small energy solutions, we establish a new critical point theorem which generalize the dual Fountain Theorem of Bartsch and Willen, by using the index theory and the $$\mathcal {P}$$ -topology. Some non-periodic conditions on the whole space $$\mathbb {R}^{3}$$ are given in order to overcome the lack of compactness.