Some hemivariational inequalities in the Euclidean space

The purpose of this paper is to study the existence of weak solutions for some classes of hemivari- ational problems in the Euclidean space.These hemivariational inequalities have a variational structure and, thanks to this, we are able to nd a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz con- tinuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group and their actions on the classical Sobolev space. Moreover, under an additional hypotheses on the dimension d and in the pres- ence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with dierent symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.