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.

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We should like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement. The work was supported by the National Science Foundation of China (NSFC11871242).

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Correspondence to Xiaojing Dong.

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Ding, Y., Dong, X. Infinitely many solutions of Dirac equations with concave and convex nonlinearities. Z. Angew. Math. Phys. 72, 39 (2021).

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  • Dirac equation
  • Generalized dual fountain theorem
  • Concave and convex nonlinearities
  • Non-periodic potential

Mathematics Subject Classification

  • 35Q40
  • 49J35