Critical point methods. (English) Zbl 1157.35050

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.


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
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