Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential
The paper studies the boundary value problem for the second order periodic system \[ -u''(t)-A(t)u(t)\in \partial F(t,u(t))\quad \text{ a.e. } t\in [0,b],\quad u(0)=u(b),\quad u'(0)=u'(b), \] where \(F:[0,b]\times {\mathbb R}^n\to {\mathbb R}\) is a measurable function, locally Lipschitz in the second variable, \( \partial F(t,x)\) is the Clarke generalized subdifferential of the locally Lipschitz map \(F(t,.)\) and \(A(.)\) is a continuous map from \([0,b]\) into the space of symmetric \((n\times n)\) - matrices. When \(\partial F(t,.)\) has linear growth both near the origin and near infinity and the linear differential operator \(u\to -u''-A(.)u\) has a nontrivial negative spectrum it is proved that the system considered has at least two nontrivial solutions. The proof essentially uses a nonsmooth variant of the reduction method and the nonsmooth critical point theory.
scientific article; zbMATH DE number 5989670