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Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. (English) Zbl 0612.58011
After defining the notion of the critical point for the function of the type \(I=\Phi +\psi: X\to (-\infty,+\infty]\), where X is a real Banach space, \(\Phi \in C^ 1(X,R)\) and \(\psi\) : \(X\to (-\infty,+\infty]\) is a convex, proper and lower semicontinuous function, the author develops a theory of the minimax characterization of the critical points of such functions. He provides us with a number of existence theorems for critical points (the mountain pass theorem and some of its generalizations, the saddle point theorem, and certain theorems concerning the critical points of even functions). The results are then applied to certain variational inequalities with single and multivalued operators which arise in the theory of elliptic boundary value problems. In the paper the Ekeland’s variational principle and an author’s deformation result are extensively used, and the results generalize (to the functions of the above mentioned type) some theorems of Ambrosetti, Rabinowitz and Clark.
Reviewer: Z.Denkowski

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J40 Variational inequalities
35J40 Boundary value problems for higher-order elliptic equations
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