The structure of the critical set in the mountain pass theorem

Authors:
Patrizia Pucci and James Serrin

Journal:
Trans. Amer. Math. Soc. **299** (1987), 115-132

MSC:
Primary 58E05; Secondary 49F15

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869402-1

MathSciNet review:
869402

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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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869402-1

Keywords:
Critical point theory,
nonlinear functional analysis

Article copyright:
© Copyright 1987
American Mathematical Society