Best approximation of orbits in iterated function systems
Yanheng Ding , , Xiaojing Dong and Qi Guo
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
Received October 2020 Revised December 2020 Published September 2021 Early access January 2021
Fund Project: The authors are supported by NSFC 11871242, 11801545
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*} $
$ 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 $
$ \liminf\limits_{|x|\rightarrow \infty} V(x) $
, as well as
$ \max W $
$ \limsup\limits_{|x|\rightarrow \infty} W(x) $
for the subcritical case and
$ \max W_{j} $
$ \limsup\limits_{|x|\rightarrow \infty} W_{j}(x), j = 1,2, $
for the critical case.
Keywords: Nonlinear Dirac equation, Semi-classical solutions, Multiplicity, Variational methods, subcritical and critical nonlinearity.
Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.
Citation: Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030
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Yanheng Ding Xiaojing Dong Qi Guo
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