  The purpose of this paper is to study the existence of weak solutions for
some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$
($d\geq 3$). These hemivariational inequalities have a variational structure
and, thanks to this, we are able to find 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 continuous 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
$O(d)$ and their actions on the Sobolev space $H^1(\mathbb{R}^d)$. Moreover,
under an additional hypotheses on the dimension $d$ and in the presence of
symmetry on the nonlinear datum, the existence of multiple pairs of
sign-changing solutions with different symmetries structure has been proved. In
connection to classical Schr\"{o}dinger equations a concrete and meaningful
example of an application is presented.
