On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $

Yanheng Ding; Xiaojing Dong; Qi Guo

doi:10.3934/dcds.2021030

Article Contents

2021, Volume 41, Issue 9: 4105-4123. Doi: 10.3934/dcds.2021030

This issue Previous Article Best approximation of orbits in iterated function systems Next Article Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $

On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China, University of Chinese Academy of Sciences, Beijing 100049, P. R. China

^* Corresponding author: dingyh@math.ac.cn
^* Corresponding author: dingyh@math.ac.cn

Received: October 2020

Revised: December 2020

Early access: January 2021

Published: September 2021

The authors are supported by NSFC 11871242, 11801545.

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Abstract

In this paper, we study multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $:

$ \begin{equation*} -i\hbar\sum\limits_{k = 1}^n \alpha_k \partial_k u+a\beta u+V(x)u = f(x,|u|)u,\; \text{in}\ \mathbb{R}^n, \end{equation*} $

where $ n\geq 2 $, $ \hbar>0 $ is a small parameter, $ a>0 $ is a constant, and $ f $ describes the self-interaction which is either subcritical: $ W(x)|u|^{p-2} $, or critical: $ W_{1}(x)|u|^{p-2}+W_{2}(x)|u|^{2^*-2} $, with $ p\in (2,2^*), 2^* = \frac{2n}{n-1} $. The number of solutions obtained depending on the ratio of $ \min V $ and $ \liminf\limits_{|x|\rightarrow \infty} V(x) $, as well as $ \max W $ and $ \limsup\limits_{|x|\rightarrow \infty} W(x) $ for the subcritical case and $ \max W_{j} $ and $ \limsup\limits_{|x|\rightarrow \infty} W_{j}(x), j = 1,2, $ for the critical case.

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Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.

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