# American Institute of Mathematical Sciences

August  2017, 37(8): 4329-4346. doi: 10.3934/dcds.2017185

## On coupled Dirac systems

 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

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  October 2016 Revised  February 2017 Published  April 2017

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.
Citation: Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
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