Dynamical systems with effective-like potentials. (English) Zbl 0648.34050

Coti Zelati, Vittorio

Nonlinear Anal., Theory Methods Appl. 12, No. 2, 209-222 (1988).

In this paper it is studied the problem of existence of T-periodic solutions for the system of differential equations (V) \(-x''=\nabla_ xV(t,x)\) where V(t,x) is assumed to be singular in the sense that \(V(t,x)\to +\infty\) as \(| x| \to 0\). Assuming: (i) V(t,x) is T- periodic in t; (ii) V(t,x), \(\nabla_ xV(t,x)\to 0\) as \(| x| \to \infty\); (iii) V(t,\(\lambda\) x) is increasing in \(\lambda\) for \(| \lambda |\) small or large (uniformly in t) and (iv) V(t,x) satisfies a “strong force” condition (which concerns the behaviour of V(t,x) close to the singularity \(x=0)\), it is proved the existence of at least one solution. In the case V does not depend on time, it is proved that it exists \(T_ 0>0\) such that \(\forall T>T_ 0\) there exists a T-periodic, non-trivial solution. Such results are proved applying Morse theory to the functional associated to (V), i.e. to the functional \(f(x)=\int^{T}_{0}| x'|^ 2dt-\int^{T}_{0}V(t,x)dt\) and studying the topology of its level sets.

Reviewer: V.Coti Zelati

MSC:

34C25	Periodic solutions to ordinary differential equations
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

singular potentials; Morse theory

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