Abstract
In this chapter we extend the minimax characterizations of critical points of functionals considering saddle-point theorems. First we present an earlier version of a saddle-point theorem of Von Neumann [Neu] and some generalizations. The goal of this chapter is to present the saddle-point theorem of P. Rabinowitz and some extensions obtained by several authors.
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Bibliography
Ahmad S., Lazer AC, Paul JL. Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math.J. 1976;196:933–944.
Amann H. Saddle Points and Multiple Solutions of Differential Equations. Math. Zeitschr. 1979;169:127–166.
Aubin, Jean-Pierre and Ekeland, Ivar. Applied Nonlinear Analysis. N.Y.: John Wiley & Sons, 1984.
Bates P, Ekeland I. A saddle point theorem. Differential Equations, Academic Press, London, 1980.
Berge C. Sur une convexité règuliere et ses applications à la théorie des jeux. Bull. Soc. Math. France, 1954;82:301–319.
Brézis H, Nirenberg L. Remarks on finding critical points. Comm. Pure and Appl. Math., 1991;XLIV:939–963.
Chang, Kung. Infinite Dimensional Morse Theory and Multiple Solution Problems. Boston, Basel, Berlin: Birkhäuser, 1993.
Deimling Klaus. Nonlinear Functional Analysis. Berlin, Heidelberg: Springer-Verlag, 1985.
Fan K. Minimax theorems. Proc.Nat. Acad. Sci., 1953;39;42–47.
Hofer H. A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem. J. London. Math. Soc.,1985;31:566–570.
Kneser H. Sur un théorème fondamental de la théorie des jeux. C.R. Acad. Sci.,Paris 1952; 234:2418–2420.
Krasnosel’skii, Mark. Topological Methods in the Theory of Nonlinear Integral Equations, Moskow, 1956 (Russian).
Lazer AC, Landesman EM, Meyers DR. On saddle point theorem in the calculus of variations, the Ritz algorithm and monotone convergence. J. Math. Anal. Appl., 1975;52:591–614.
Lazer AC, McKenna PJ. Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. Comm. in PDE, 1985;10:107–150.
Lazer AC, Solimini S. Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Analysis, T.M.A., 1988;12:761–775.
Li ST, Willem M. Applications of local linking to critical point theory. J Math. Anal. Appl., 1995;189:6–32.
Lin JQ, List SJ. Some existence theorems on multiple critical points and their applications, Kexue Tongbao 1984; 17.
Manasevich R. A minimax theorem. J Math. Anal. App1.,1982;90:64–71.
Mawhin, Jean and Willem, Michael. Critical Point Theory and Hamiltonian Systems. N.Y.: Springer-Verlag, 1988.
Moroz V, Vignoli A, Zabreiko P. On the three critical point theorem, Topological Methods in Nonlinear Analysis, 1998;11:103–113.
Von Neumann J. Zur Theorie der Gesellschaftspiele, Math. Ann. 1928;100:295–320.
Nirenberg L. Topics in nonlinear functional analysis, Courant Inst. Math. Sci., New York University, NY, 1974.
Rabinowitz P. Minimax methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. 65, AMS, Providence, R.I., 1986.
Rabinowitz P. Some minimax theorems and applications to nonlinear partial differential equations. Nonlinear analysis, A collection of papers in honour of Eric Rothe. Academic Press, N.Y., 1978:161–177.
Ramos, Miguel. Teorermas de Enlace na Teoria dos Pontos Críticos. Universidade de Lisboa, Faculdade de Ciências, 1993.
Schechter M. A generalization of the saddle point method with applications. Anales Polonici Mathematici, 1992; LVII.3:269–281.
Schechter M. New saddle point theorems. Generalized functions and their applications, Banaras Hindu University, 1991.
Schwartz John, Nonlinear Functional Analysis. Gordon and Breach, New York, 1969.
Shiffman. On the equality minmax-maxmin, and the theory of games, RAND, Report RM - 243, 1949.
Silva EA. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Analysis, TMA, 1991;16:455–477.
Smoler Joel. Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, Heidelberg, 1983.
Solmini S. Existence of a third solution for a class of BVP with jumping nonlinearities. Nonlinear Analysis, TMA 1983;7:917–927.
Tersian S. A minimax theorem and applications to nonresonance problems for semilinear equations. Nonlinear Analysis, T.M.A.,1986:10:651–688.
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do Rosário Grossinho, M., Tersian, S.A. (2001). Saddle-Point and Linking Theorems. In: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Nonconvex Optimization and Its Applications, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3308-2_2
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DOI: https://doi.org/10.1007/978-1-4757-3308-2_2
Publisher Name: Springer, Boston, MA
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