Question

The lusternik-schnirelman method relates the topology of manifolds with the critical points of functionals defined on them, giving lower bounds for the number of critical points in terms of the lusternik-schnirelmann category.

Is there some reference for the lusternik-schnirelman method (ensuring existence of at least cat(x) critical points for a functional defined on a banach manifold (x), in the context of quantitative deformation, as understood by Willem?).

The functional is not asumed to satisfy palais-smale condition (guaranteed by a deformation property with respect to compact critical sets), but rather a quantitative deformation lemma with respect to possibly noncompact critical sets.

Answer 1

I happened to wirte something about that, in old times. As explained in the introduction and in the last section, the idea is that in the application one should consider the given functional on a suitable submanifold with boundary, where a corresponding form of the Palais-Smale condition holds, and apply a Lusternik Schnirel'man theory there.

http://www.maths.ed.ac.uk/~aar/topology/Topology%20Vol%2034/Majer_Two-variational-methods-on-manifolds-with-boundary_1995.pdf