## 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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