# American Institute of Mathematical Sciences

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On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations
January  2020, 19(1): 587-607. doi: 10.3934/cpaa.2020028

## Potential well and multiplicity of solutions for nonlinear Dirac equations

 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex System, Ministry of Education, 100875 Beijing, China 2 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, China 3 Center for Applied Mathematics, Tianjin University, 300072 Tianjin, China

Received  September 2018 Revised  September 2018 Published  July 2019

In this paper we consider the semi-classical solutions of a massive Dirac equations in presence of a critical growth nonlinearity
 $-i\hbar \sum\limits_{k = 1}^{3}\alpha_k\partial_k w+a\beta w+V(x)w = f(|w|)w.$
Under a local condition imposed on the potential
 $V$
, we relate the number of solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.
Citation: Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028
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