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 \(V\).


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