Hemivariational inequalities with the potential crossing the first eigenvalue. (Q2777714)

Let \(\Omega\) be a smooth, bounded domain in \({\mathbb R}^N\) and \(2\leq p<\infty\). The present paper is concerned with the quasilinear hemivariational inequality \(-\text{ div}\, (| | Dx(z)| | ^{p-2}Dx(z))\in \partial j(z,x(z))\) in \(\Omega\), subject to the Dirichlet boundary condition \(x=0\) on \(\partial\Omega\), where \(j:\Omega\times {\mathbb R}\rightarrow {\mathbb R}\) is a Carathéodory function such that \(j(z,\cdot )\) is locally Lipschitz, for any \(z\in\Omega\). The potential \(j\) is assumed to be subcritical and to have an asymptotic behaviour which goes beyond the first eigenvalue of the \(p\)-Laplace operator in \(W_0^{1,p}(\Omega)\). Using a nonsmooth version of the Linking Theorem the authors establish the existence of at least a solution to the above problem. The proof of this result is based on a careful analysis of the corresponding energy functional, combined with specific methods employed in the nonsmooth critical point theory.