On coupled Dirac systems

Abstract

In this paper, we show the existence of solutions for the coupled Dirac system \begin{document}$\left\{ \begin{aligned}Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\\Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on}\hspace{2mm}M,\end{aligned} \right.$\end{document} 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 \begin{document}$H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1},$\end{document} where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy \begin{document}$\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}.$\end{document} In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.