Some existence results for a periodic problem with non-smooth potential (Q984101)

The authors study the boundary value problem \[ -u''(t)+g(t)\in \partial j(t,u(t)), \quad t\in (0,2\pi ),\quad u(0)=u(2\pi ),\quad u'(0)=u'(2\pi ),\tag{1} \] where \(g(.)\) is a bounded measurable real function, \(j(.,.)\) is locally Lipschitz in the second variable and \(\partial j(.,.)\) is Clarke's subdifferential of the function \(j(.,.)\). The paper contains two main results. If the potential function \(j(.,.)\) is not smooth in the second variable, the authors prove under certain hypotheses that problem (1) has a solution. The proof is based on the so-called ``reduction method'' developed in the smooth case and adapted to problem (1). In the second result, it is proved that problem (1) has at least two distinct nontrivial solutions. This is done by using a certain non-smooth critical point theorem and a recent multiplicity result based on local linking.