Critical point methods. (English) Zbl 1157.35050

Schechter, Martin

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 3, 987-999 (2008).

The author presents a survey of abstract approaches to the problem of existence of critical points of a \(C^1\)-functional \(G\) defined on a Banach space \(E\). More precisely, he explains how one obtains a Cerami sequence for \(G\) in various settings. A Cerami sequence for \(G\) is a sequence \((u_n)\subseteq E\) such that, for some \(c\in\mathbb{R}\), \(G(u_n)\to c\) and \((1+\| u_n\| )\| G'(u_n)\| \to 0\). In relevant examples, this leads to the existence of a critical point of \(G\) with functional value \(c\). The use of Cerami sequences instead of Palais-Smale sequences poses no restrictions, but gives better results in the applications.
At first, the author gives a brief review of his recent results on sandwich Pairs. Then he develops min-max theorems from an abstract point of view, and explains some applications to semilinear elliptic boundary value problems.

Reviewer: Nils Ackermann (México)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J35	Existence of solutions for minimax problems
35J20	Variational methods for second-order elliptic equations

Keywords:

critical point theory; Palais-Smale sequence; Cerami sequence; sandwich pair; min-max theorem

References:

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