Interfaces in some elliptic nonlinear boundary value problems on Riemannian manifolds: Necessary condition and location

We show that the equal-area condition is a necessary hypothesis for the existence of layered solutions to a nonlinear boundary value problem depending on a positive real parameter on a Riemannian manifold with or without boundary. As an application we give examples where the results can be used in order to determine in advance the location of possible limiting interface as the parameter goes to zero.