A topological approach to hemivariational inequalities with unilateral growth condition (Q2747066)

The authors consider the following hemivariational inequality on a vector-valued function space: NEWLINE\[NEWLINEa(u,v- u)+ \int_\Omega j^0(x,u; v-u) dx\geq \langle g,v- u\rangle_V,\quad\forall v\in V,NEWLINE\]NEWLINE where \(V\) is a Banach space compactly imbedded in \(L^p(\Omega; \mathbb{R}^N)\) \((p> 2)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^m\) with sufficiently smooth boundary and \(j^0\) denotes the Clarke directional derivative of a locally Lipschitz function.NEWLINENEWLINENEWLINEAssuming a unilateral growth condition for the nonlinear nonconvex part, an existence result for the above hemivariational inequality is shown. To this end, the critical point theory and the Galerkin approximation method is applied. The authors exploit the nonsmooth version of the Mountain Pass Theorem by [\textit{K.-C. Chang}, J. Math. Anal. Appl. 80, 102-129 (1981; Zbl 0487.49027)] and some a priori estimates for the finite-dimensional approximate solutions on the basis of their minimax characterizations and finally a process of passing to the limit.NEWLINENEWLINENEWLINEThe paper improves substantially results presented in [\textit{D. Motreanu} and \textit{Z. Naniewicz}, Differ. Integral Equ. 9, No. 3, 581-598 (1996; Zbl 0848.35051)].