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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
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: http://dx.doi.org/10.1090/S0002-9947-1987-0869402-1
PII: S 0002-9947(1987)0869402-1
Keywords: Critical point theory, nonlinear functional analysis
Article copyright: © Copyright 1987 American Mathematical Society