Existence results for solutions to nonlinear Dirac systems on compact spin manifolds (Q680358)

The paper deals with existence of solutions to the following Dirac system \[ \begin{cases} Du=\dfrac{\partial H}{\partial v}(x,u,v)\quad & \text{on}\;M,\\ Dv=\dfrac{\partial H}{\partial u}(x,u,v)\quad & \text{on}\;M. \end{cases} \] Here \((M,g)\) is an \(m\)-dimensional, \(m\geq2,\) compact oriented Riemannian manifold equipped with a spin structure \(\rho: P_{\text{spin}(M)}\to P_{\text{so}(M)},\) \(\Sigma M=P_{\text{spin}(M)}\times_\sigma \Sigma_m\) stands for the complex spinor bundle on \(M,\) \(u,v\in C^\infty(M,\Sigma M)\) are spinors, \(D\) is the Dirac operator on \(M,\) while \(H:\;\Sigma M\oplus \Sigma M\to \mathbb{R}\) is a real-valued superquadratic \(C^1\)-function with subcritical growth rates. The author proves existence of nontrivial solutions to the Dirac system by combining Galerkin-type approximations with linking arguments.