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Periodic solutions for second order systems with not uniformly coercive potential. (English) Zbl 0999.34039

The authors study the second-order system \[ \ddot u(t)=\nabla F(t,u(t))\quad \text{a.e.}\quad t\in[0, T],\qquad u(0)-u(T)=\dot u(0)-\dot u(T)=0, \] with locally coercive potential, that is \(F(t,x)\rightarrow\infty\) a.e. for \(t\) in some positive measure subset of \([0, T]\). Existence and multiplicity of periodic solutions are obtained. The result is established using an analogy of Egorov’s theorem, properties of subadditive functions, the least action principle, and a three-critical-point theorem proposed by Brezis and Nirenberg.

MSC:

34C25 Periodic solutions to ordinary differential equations
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