Positive solutions of singularly perturbed nonlinear elliptic equations on Riemannian manifolds with boundary

Let (M,g) be a smooth connected compact Riemannian ma- nifold of finite dimension n ≥ 2 with a smooth boundary. We consider a singularly perturbed elliptic equation on M with homogeneous Neumann boundary condition. The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.