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.
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$\begingroup$ Mountain pass theorem finds saddle points. What are you precisely looking, a minimum of the functional or a saddle point? $\endgroup$– TommiFeb 5, 2015 at 7:59

1$\begingroup$ There are generalisations of the theorem: see for example references [17] and [24] in master's thesis of Eero Ruosteenoja: jyx.jyu.fi/dspace/handle/123456789/41049 $\endgroup$– TommiFeb 5, 2015 at 8:25

$\begingroup$ Roughly speaking, consider a constrained minimization problem with two distinct minimizers. Can we generalize the mountain pass theorem to find a third critical point (a saddle point) for the constrained problem? $\endgroup$– A random mathematicianFeb 5, 2015 at 10:05

1$\begingroup$ Linking theorem might be useful. The aforementioned references should contain relevant material. $\endgroup$– TommiFeb 5, 2015 at 13:04
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1 Answer
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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.