# Quantitative deformation and lusternik schnirelman method

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.

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(PS: due to a curious typo in the first lines the manifold $M$ is also denoted $N$) –  Pietro Majer Jul 24 '12 at 7:27