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


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
Full Text: DOI


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