In this thesis we study the existence and multiple solutions of the nonlinear Dirac equation (0.0.1) and the nonlinear coupled Dirac system (0.0.2) with a potential on a compact spin manifold.

In chapter 2, using the saddle point reduction we obtain an existence result and a multiple one about nontrivial solutions of (0.0.1).

In chapter 3, we firstly study the spectrum of operator L and then prove an existence result and an infinitely many multiple one about nontrivial solutions of (0.0.2) via Galerkin type approximations and linking arguments.

In chapter 4, we consider (0.0.1) and (0.0.2) with even nonlinear terms. If all eigenvalues of D are nonzero, using an recent generalization of Clark’s theorem we prove that both (0.0.1) and (0.0.2) possess infinitely many solutions.