On Coupled Dirac Systems under Boundary Condition

In this article we study the existence of solutions for the Dirac systems \begin{equation}\label{e:0.1} \left{ \begin{array}{c} Pu=\frac{\partial H}{\partial v}(x,u,v) \quad\hbox{on} \ M, Pv=\frac{\partial H}{\partial u}(x,u,v) \quad\hbox{on} \ M, B_{$\text{CHI}$}u= B_{$\text{CHI}$}v=0\quad\hbox{on} \ \partial M \end{array} \right. \end{equation} where $M$ is an $m$-dimensional compact oriented Riemannian spin manifold with smooth boundary $\mathrm{\partial }M$, $P$ is the Dirac operator under the boundary condition $B_{\text{CHI}}u=B_{\text{CHI}}v=0$ on $\mathrm{\partial }M$, $u,v\in C^{\mathrm{\infty }}(M,\mathrm{\Sigma }M)$ are spinors. Using an analytic framework of proper products of fractional Sobolev spaces, the solutions existence results of the coupled Dirac systems are obtained for nonlinearity with superquadratic growth rates.