## 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 $$\Phi''(x)$$ 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 $$H_ q(\{f\leq b\},\{f\leq a\})\approx 0\forall q$$. 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 $$\dot z=JH_ z(z,t)$$ with asymptotically linear Hamiltonian.
Reviewer: A.Szulkin

### MSC:

 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 34C25 Periodic solutions to ordinary differential equations 55N99 Homology and cohomology theories in algebraic topology 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Zbl 0275.55009
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### References:

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