# Some hemivariational inequalities in the Euclidean space

## TL;DR

This paper investigates the existence of weak solutions for certain hemivariational inequalities in Euclidean space using variational methods and symmetry principles, also demonstrating multiple solutions with different symmetries.

## Contribution

It introduces a variational approach combined with a non-smooth criticality principle to establish existence and multiplicity of solutions for hemivariational problems in high-dimensional spaces.

## Key findings

- Existence of at least one weak solution established.
- Multiple sign-changing solutions with distinct symmetries proved.
- Application to classical Schrödinger equations demonstrated.

## Abstract

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.