Potential well and multiplicity of solutions for nonlinear Dirac equations

Yu Chen; Yanheng Ding; Tian Xu

doi:10.3934/cpaa.2020028

Article Contents

2020, Volume 19, Issue 1: 587-607. Doi: 10.3934/cpaa.2020028

This issue Previous Article Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method Next Article On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations

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: August 31, 2018

Revised: August 31, 2018

Published: July 2019

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Abstract

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.

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Mathematics Subject Classification: 35Q40, 49J35.

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