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Periodic solutions for second order Hamiltonian systems with a change sign potential. (English) Zbl 1078.34023

The authors deal with the following second-order Hamiltonian system \[ \begin{gathered} u''(t)+ b(t)\nabla V(u(t))= 0\quad\text{a.e. }t\i [0,T],\\ u(0)- u(T)= \dot u(0)- \dot u(T)= 0,\end{gathered}\tag{1} \] with \(T> 0\), \(b\in C([0,T],\mathbb{R})\) and \(V\in C^1(\mathbb{R}^N, \mathbb{R})\). In contrast to many papers, the authors assume that \(b\) changes sign and \(\int^T_0 b(t)\,dt= 0\). Using the minimax methods in critical point theory, the authors prove the existence of at least one nonzero solution of (1) under suitable assumptions on \(V\).

MSC:

34C25 Periodic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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