Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see <http://en.wikipedia.org/wiki/Mountain_pass_theorem> for the precise statement of the theorem). I wonder if there is a generalization of this theorem for minimization problems with constraints. 



Mountain Pass theorem for minimization problems with constraints
You can generalize the mountain pass theorem if the constraints consist of a Banach manifold.
This is because you can construct a pseudo gradient flow on a Banach manifold, which is required in the proof of the theorem.
Check Chapter 27 of 'Nonlinear Functional Analysis' by K. Deimling.


For example, in the paper 'Standing waves of some coupled nonlinear Schroedinger equations' by A. Ambrosetti and E. Colorado (2007), the authors used the mountain pass theorem on a Nehari manifold.


