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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.

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
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References:

[1] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. pure appl. math., 44, 939-964, (1991) · Zbl 0751.58006
[2] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag · Zbl 0676.58017
[3] Schechter, M., New linking theorems, Rend. sem. mat. univ. Padova, 99, 255-269, (1998) · Zbl 0907.35053
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