The structure of the critical set in the mountain pass theorem

Pucci, Patrizia; Serrin, James

doi:10.1090/S0002-9947-1987-0869402-1

The structure of the critical set in the mountain pass theorem
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by Patrizia Pucci and James Serrin PDF

Trans. Amer. Math. Soc. 299 (1987), 115-132 Request permission

Abstract:

We show that the critical set generated by the Mountain Pass Theorem of Ambrosetti and Rabinowitz must have a well-defined structure. In particular, if the underlying Banach space is infinite dimensional then either the critical set contains a saddle point of mountain-pass type, or the set of local minima intersects at least two components of the set of saddle points. Related conclusions are also established for the finite dimensional case, and when other special conditions are assumed. Throughout the paper, no hypotheses of nondegeneracy are required on the critical set.

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