Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

Let (M,g)(M,g) be a smooth, compact Riemannian manifold with smooth boundary, with n=dimM=2,3n=dimM=2,3. We suppose the boundary ∂M∂M to be a smooth submanifold of MM with dimension n−1n−1. We consider a singularly perturbed nonlinear system, namely Klein–Gordon–Maxwell–Proca system, or Klein–Gordon–Maxwell system of Schroedinger–Maxwell system on MM. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary ∂M∂M, by means of the Lusternik–Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.