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Periodic solutions of dynamical systems by a saddle point theorem of Rabinowitz. (English) Zbl 0729.58044
The author considers the problem of finding periodic solutions of differential equations of the type \[ (1)\quad \ddot x+\nabla_ xV(t,x)=0\quad (x\in {\mathbb{R}}^ n,\quad V\in C^ 1({\mathbb{R}}\times {\mathbb{R}}^ n,{\mathbb{R}}),\quad V(t+T,x)=V(t,x)). \] Using some special cases of linking techniques [see, for instance, V. Benci and P. H. Rabinowitz, Invent. Math. 52, 241-273 (1979; Zbl 0465.49006) and P. H. Rabinowitz, Lect. Notes Math. 648, 97-115 (1978; Zbl 0377.35020)] he establishes some sufficient conditions for periodic solutions to exist:
Theorem 1. If V(t,x)\(\leq 0\forall (t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n\), \(V\not\equiv 0\), V(t,x)\(\to 0\) and \(\nabla_ xV(t,x)\to 0\) as \(\| x\| \to \infty\) uniformly in t, then equation (1) has at least one T- periodic solution x(t) such that \(V(t,x(t))<0\) for some t and there exists a sequence of distinct subharmonic solutions.
Theorem 2. If \(V(t,x)\to +\infty\) as \(\| x\| \to \infty\) uniformly in t and \(\nabla_ xV(t,x)\) is bounded, then for every \(k\in N\) there exists a kT-periodic solution \(x_ k(t)\) such that \(\| x_ k\|_{L\infty}\to +\infty\) as \(k\to +\infty.\)
A generalization of Theorem 1, in which V(t,x) is bounded and may change its sign is considered, too.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
35B10 Periodic solutions to PDEs
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