# zbMATH — the first resource for mathematics

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
Full Text:
##### 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. P.; Ekeland, I., Applied nonlinear analysis, (1984), Wiley New York [3] Barbu, Y., Nonlinear semigroups and differential equations in Banach spaces, (1976), Editura Academiei. Bucarest and Nordhoff Leyden [4] Brézis, H., Opérateurs-maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1973), North-Holland Amsterdam · Zbl 0252.47055 [5] Brézis, H., Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations, (Zarantanello, E., Contributions to Nonlinear Functional Analaysis, (1971), Academic Press New York), 101-156 [6] H. Brézis, Some variational problems with lack of compactness, Proc. of the 1983 AMS Summer Institute on Nonl. Func. Anal. and Appl., Amer. Math. Soc. (to appear). [7] Brézis, H.; Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa, t. 5, Ser. IV, 225-326, (1978) · Zbl 0386.47035 [8] Chang, K. 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) [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 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. E., t. 9, 699-717, (1984) · Zbl 0552.35030 [15] Hedberg, L. I., Spectral synthesis and stability in Sobolev spaces, Springer Lecture Notes in Mathematics, t. 779, 73-103, (1980) [16] Hu, S. T., Homotopy theory, (1959), Academic Press New York · Zbl 0041.51902 [17] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (1980), Academic Press New York · Zbl 0457.35001 [18] Kuratowski, K., Topologie, (1958), PWN Warsaw · JFM 57.0758.05 [19] Mawhin, J.; Willem, M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Diff. Eq., t. 52, 264-287, (1984) · Zbl 0557.34036 [20] Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc., t. 4, 267-302, (1981) · Zbl 0468.47040 [21] Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, proc. sym. on eigemvalues of nonlinear problems, 143-195, (1974), 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 New York), 161-177 [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 [24] P. H. Rabinowitz, Some aspects of critical point theory, MRC Tech. Rep. ≠ 2465, Madison, Wisconsin, 1983. [25] Schwartz, J. T., Nonlinear functional analysis, (1969), 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 [27] M. Struwe: Generalized Palais-Smale conditions and applications. Vorlesungsreihe SFB #17, Bonn, 1983. · Zbl 0534.58021 [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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.