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Cohomology and Morse theory for strongly indefinite functionals. (English) Zbl 0735.58012
Let ${\Phi }$ be a functional which is strongly indefinite in the sense that ${{\Phi }}^{\text{'}\text{'}}\left(x\right)$ has infinitely many positive and infinitely many negative eigenvalues. For such ${\Phi }$ the usual (co)homology and Morse theories cannot be employed because all critical points have infinite Morse index and ${H}_{q}\left(\left\{f\le b\right\},\left\{f\le a\right\}\right)\approx 0\forall q$. In this paper an infinite dimensional cohomology theory (of Gȩba-Granas type, cf. K. Gȩba and A. Granas, J. Math. Pure Appl., IX. Ser. 52, 145- 270 (1973; Zbl 0275.55009)) is constructed. It satisfies all the Eilenberg-Steenrod axioms except the dimension axiom which is quite different (certain infinite dimensional spheres have nontrivial cohomology). This cohomology gives rise to a Morse theory which turns out to be useful for studying the number of critical points of strongly indefinite functionals. The abstract theory is applied to the problem of existence of multiple time-periodic solutions for Hamiltonian systems $\stackrel{˙}{z}=J{H}_{z}\left(z,t\right)$ with asymptotically linear Hamiltonian.
Reviewer: A.Szulkin
##### MSC:
 58E05 Abstract critical point theory 34C25 Periodic solutions of ODE 55N99 Homology and cohomology theories (algebraic topology) 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
##### References:
 [1] Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Sc. Norm. Super. Pisa, CC. Sci., IV. Ser.7, 539–603 (1980) [2] Amann, H., Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian systems. Manuscr. Math.32, 149–189 (1980) · Zbl 0443.70019 · doi:10.1007/BF01298187 [3] Benci, V.: A new approach to the Morse-Conley theory. In: Dell’Antonio, G.F., D’Onofrio, B. (eds.) Proc. Int. Conf. on Recent Advances in Hamiltonian Systems, pp. 1–52 Singapore: World Sci. Publishing 1987 [4] Chang, K.C.: Infinite Dimensional Morse Theory and Its Applications. (Sémin. Math. Supér., vol. 97) Montréal: Les Presses de l’Université de Montréal 1985 [5] Conley, C.C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math.73, 33–49 (1983) · Zbl 0516.58017 · doi:10.1007/BF01393824 [6] Dancer, E.N.: Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math.350, 1–22 (1984) · doi:10.1515/crll.1984.350.1 [7] Dold, A.: Lectures on Algebraic Topology. Berlin Heidelberg New York: Springer 1972 [8] Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princton, N.J.: Princeton University Press 1952 [9] Fonda, A., Mawhin, J.: Multiple periodic solutions of conservative systems with periodic nonlinearity. (Preprint) [10] Gęba, K., Granas, A.: Algebraic topology in linear normed spaces I–V. Bull. Acad. Pol. Sci. I:13, 287–290 (1965); II:13, 341–346 (1965); III:15, 137–143 (1967); IV:15, 145–152 (1967); V:17, 123–130 (1969) [11] Gęba, K., Granas, A.: Infinite dimensional cohomology theories. J. Math. Pure Appl.52, 145–270 (1973) [12] Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology8, 361–369 (1969) · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6 [13] Li, S., Liu, J.Q.: Morse theory and asymptotic linear Hamiltonian system. J. Differ. Equations78, 53–73 (1989) · Zbl 0672.34037 · doi:10.1016/0022-0396(89)90075-2 [14] Liu, J.Q.: A generalized saddle point theorem. J. Differ. Equations82, 372–385 (1989) · Zbl 0682.34032 · doi:10.1016/0022-0396(89)90139-3 [15] Mardešić, S., Segal, J.: Shape Theory. Amsterdam: North-Holland 1982 [16] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Berlin Heidelberg New York: Springer 1989 [17] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. (Reg. Conf. Ser. Math., vol. 65) Providence, R.I.: Am. Math. Soc. 1986 [18] Spanier, E.H.: Algebraic Topology New York: McGraw-Hill, 1966 [19] Szulkin, A.: A relative category and applications to critical point theory for strongly indefinite functionals. J. Nonl. Anal.15, 725–739 (1990) · Zbl 0719.58011 · doi:10.1016/0362-546X(90)90089-Y