On coupled Dirac systems

Wenmin Gong; Guangcun Lu

doi:10.3934/dcds.2017185

Article Contents

2017, Volume 37, Issue 8: 4329-4346. Doi: 10.3934/dcds.2017185

This issue Previous Article Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one Next Article Statistical and deterministic dynamics of maps with memory

On coupled Dirac systems

Wenmin Gong^1, , and
Guangcun Lu^2,

1.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

^* Corresponding author: Wenmin Gong
^* Corresponding author: Wenmin Gong

The first author is supported by the NSF (grant no. 11571194) of China.
The second author is Partially supported by the NNSF (grant no. 10971014 and 11271044) of China.

Received: September 30, 2016

Revised: January 31, 2017

Published: May 2017

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Abstract

In this paper, we show the existence of solutions for the coupled Dirac system

$\left\{ \begin{aligned}Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\\Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\end{aligned} \right.$

where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the $\mathbb{Z}_2$-invariant $H$ that includes a nonlinearity of the form

$H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1},$

where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy

$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}.$

In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.

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Mathematics Subject Classification: Primary: 53C27, 57R58, 58E05, 58J05.

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