Differential inclusions involving oscillatory terms

Abstract

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))\ \mbox{in}\ \Omega \newline u\geq 0\ \mbox{in}\ \Omega \newline u= 0\ \mbox{on}\ \partial\Omega, \end{cases} \ \ \ \ \ \ \ \ \ \ \ \ {({\mathcal D}_\lambda)} \] where $\Omega \subset {\mathbb R}^n$ is a bounded open domain, and $\partial F$ and $\partial G$ stand for the generalized gradients of the locally Lipschitz functions $F$ and $G$. In this paper we provide a quite complete picture on the number of solutions of $({\mathcal D}_\lambda)$ whenever $\partial F$ oscillates near the origin/infinity and $\partial G$ is a generic perturbation of order $p>0$ at the origin/infinity, respectively. Our results extend in several aspects those of Krist\'aly and Moro\c{s}anu [J. Math. Pures Appl., 2010].