Existence Results for Solutions to Nonlinear Dirac Systems on Compact Spin Manifolds

Abstract

Abstract In this article, we study the existence of solutions for the Dirac system { D ⁢ u = ∂ ⁡ H ∂ ⁡ v ⁢ ( x , u , v ) on ⁢ M , D ⁢ v = ∂ ⁡ H ∂ ⁡ u ⁢ ( x , u , v ) on ⁢ M , \left\{\begin{aligned} \displaystyle Du&\displaystyle=\frac{\partial H}{% \partial v}(x,u,v)\quad\text{on }M,\\ \displaystyle Dv&\displaystyle=\frac{\partial H}{\partial u}(x,u,v)\quad\text{% on }M,\end{aligned}\right. where M is an m-dimensional compact Riemannian spin manifold, u , v ∈ C ∞ ⁢ ( M , Σ ⁢ M ) {u,v\in C^{\infty}(M,\Sigma M)} are spinors, D is the Dirac operator on M, and the fiber preserving map H : Σ ⁢ M ⊕ Σ ⁢ M → ℝ {H:\Sigma M\oplus\Sigma M\rightarrow\mathbb{R}} is a real-valued superquadratic function of class C 1 {C^{1}} with subcritical growth rates. Two existence results of nontrivial solutions are obtained via Galerkin-type approximations and linking arguments.