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

 

Sandwich pairs in critical point theory


Author: Martin Schechter
Journal: Trans. Amer. Math. Soc. 360 (2008), 2811-2823
MSC (2000): Primary 35J65, 58E05, 49J35
Published electronically: January 25, 2008
MathSciNet review: 2379776
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Abstract: Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional $ G$ is semibounded, one can find a Palais-Smale (PS) sequence

$\displaystyle G(u_k) \to a,\quad G'(u_k)\to 0. $

These sequences produce critical points if they have convergent subsequences (i.e., if $ G$ satisfies the PS condition). However, there is no clear method of finding critical points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for $ C^1$ functionals $ G$ that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.


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Additional Information

Martin Schechter
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: mschecht@math.uci.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04470-X
PII: S 0002-9947(08)04470-X
Keywords: Critical point theory, variational methods, saddle point theory, semilinear differential equations.
Received by editor(s): August 14, 2005
Published electronically: January 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society