Szulkin, Andrzej Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. (English) Zbl 0612.58011 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 77-109 (1986). 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 Cited in 12 ReviewsCited in 191 Documents 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 Keywords:minimax principles; Lusternik-Schnirelman theory; convex functions; critical points; Ekeland’s variational principle × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Func. Anal., t. 14, 349-381 (1973) · Zbl 0273.49063 [2] Aubin, J. 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C., Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., t. 80, 102-129 (1981) · Zbl 0487.49027 [9] Clark, D. C., A variant ofthe Ljusternik-Schnirelmann theory, Indiana Univ. Math. J., t. 22, 65-74 (1972) · Zbl 0228.58006 [10] Dias, J. P., Variational inequalities and eigenvalue problems for nonlinear maximal monotone operators in a Hilbert space, Amer. J. Math., t. 97, 905-914 (1975) · Zbl 0319.47040 [11] Dias, J. P.; Hernández, J., A Sturm-Liouville theorem for some odd multivalued maps, Proc. Amer. Math. Soc., t. 53, 72-74 (1975) · Zbl 0285.47037 [12] Duvaut, G.; Lions, J. L., Inequalities in Mechanics and Physics (1976), Springer: Springer Berlin · Zbl 0331.35002 [13] Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., t. 1, 443-474 (1979) · Zbl 0441.49011 [14] de Figueiredo, D. G.; Solimini, S., A variational approach to superlinear elliptic problems, Comm. P. D. 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Sym. on Eigemvalues of Nonlinear Problems, 143-195 (1974), Edizioni Cremonese: Edizioni Cremonese Rome [22] Rabinowitz, P. H., Some minimax theorems and applications to nonlinear partial differential equations, (Cesari, L.; Kannan, R.; Weinberger, H. F., Nonlinear Analysis, A collection of papers in honor of E. Rothe (1978), Academic Press: Academic Press New York), 161-177 · Zbl 0466.58015 [23] Rabinowitz, P. H., Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, t. 5, Ser. IV, 215-223 (1978) · Zbl 0375.35026 [25] Schwartz, J. T., Nonlinear Functional Analysis (1969), Gordon and Breach: Gordon and Breach New York · Zbl 0203.14501 [26] Struwe, M., Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., t. 261, 399-412 (1982) · Zbl 0506.35034 [28] Ekeland, I.; Lasry, J. M., On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. Math., t. 112, 283-319 (1980) · Zbl 0449.70014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.