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Periodic solutions of second order nonautonomous systems with the potentials changing sign. (English) Zbl 0799.58064

This paper is concerned with the following second order system \[ \ddot x + b(t) V'(x) = 0,\quad \text{for\;}x \in \mathbb{R}^ N, \] where \(V\in C^ 2(\mathbb{R}^ N,\mathbb{R})\) satisfies some superquadratic growth assumptions and \(b\) is a \(T\)-periodic continuous real function. Some existence results for periodic solutions of the above system are presented in Theorems 1 and 2 with positive and negative average values for \(b\), respectively. The proofs are based on the well-known techniques of critical point theory. The multiplicity results are given in Theorems 3 and 4, based on the consideration of the Nehari’s manifold associated with the functional considered.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34C25 Periodic solutions to ordinary differential equations
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