Existence and multiple solutions for nonautonomous second order systems with nonsmooth potentials (Q903956)

The authors study the following periodic problem in \(\mathbb{R}^N\) \[ \begin{aligned} {\ddot{u}}(t) \in \partial F(t,u(t)) \text{ for a.e. }t \in [0,T],\\ u(0) - u(T) = \dot{u}(0) = \dot{u}(T), \end{aligned} \] where \(\partial F\) denotes the Clarke subdifferential and the function \(F: [0,T] \times \mathbb{R}^N \to \mathbb{R}\) can be represented in the form \(F = F_1 + F_2\) with \(F_1, F_2\) integrable in the first argument, \(F_1\) is strongly differentiable and \(F_2\) is locally Lipschitz continuous in the second argument. Some results on the existence of an optimal solution and at least three distinct solutions are presented.