Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary. (June 2015) - Ghimenti, Marco; Micheletti, Anna Maria
Abstract: Let ( M, g ) be a smooth, compact Riemannian manifold with smooth boundary, with n = dim M = 2, 3 . We suppose the boundary ∂ M to be a smooth submanifold of M with dimension n − 1 . We consider a singularly perturbed nonlinear system, namely Klein–Gordon–Maxwell–Proca system, or Klein–Gordon–Maxwell system of Schroedinger–Maxwell system on M . We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary ∂ M, by means of the Lusternik–Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.