## Multiple solutions of perturbed superquadratic second order Hamiltonian systems.(English)Zbl 0676.34026

This interesting paper deals with the Hamiltonian system $$\ddot q+\text{grad} V(q)=f(t),$$ where $$f\in L^ 2([0,T],{\mathbb{R}}^ n)$$ is a T- periodic function and $$V\in C^ 1({\mathbb{R}}^ n,{\mathbb{R}})$$ fulfils the following condition: $$0<\mu V(q)\leq q\cdot \text{grad} V(q)$$ for $$\| q\| \geq r_ 0>0$$ and some $$\mu >2$$. For such system the author proves the existence of infinitely many distinct T-periodic solutions. These solutions are obtained as the critical points of certain functional J, introduced by P. Rabinowitz [(*) Trans. Am. Math. Soc. 272, 753-769 (1982; Zbl 0589.35004)]. The author follows the basic functional framework of (*), but he modifies the treatment of $$S^ 1$$-action and derives some new a priori estimates, which allows to prove the above mentioned result.
Reviewer: A.Klič

### MSC:

 34C25 Periodic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 70H05 Hamilton’s equations

Zbl 0589.35004
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### References:

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