Differential inclusions involving oscillatory terms

Kristály, Alexandru and Mezei, Ildikó-Ilona and Szilak, Karoly (2020) Differential inclusions involving oscillatory terms. Nonlinear Analysis: Theory, Methods and Applications. ISSN 0362-546X

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Abstract

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \ \left\{ \begin{array}{lll} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))& {\rm in} & \Omega; \\ u\geq 0, &\mbox{in} & \Omega;\\ u= 0, &{\rm on}& \partial \Omega, \end{array}\right. \eqno{({\mathcal D}_\lambda)}\] 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 sanu [\textit{J. Math. Pures Appl}., 2010].

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