Location of solutions to eigenvalue problems for hemivariational inequalities (Q2724674)

The author establishes the existence of eigensolutions \((u,\lambda)\) of some nonlinear eigenvalue problems as well as the location of these eigensolutions by means of the graph of a suitably chosen function. As the model problem the following hemivariational inequality is considered: Find \(u\in V\setminus\{0\}\) and \(\lambda\in\mathbb{R}\) such that NEWLINE\[NEWLINE\begin{cases}\langle Au,v\rangle_{V^*\times V}\leq \lambda[\int_\Omega j^0(x,u(x); v(x)) dx+\langle f,v\rangle_{V^*\times V}]\;\forall v\in V, \\ 0<\lambda<\overline\lambda\end{cases} NEWLINE\]NEWLINE where \(\Omega\) is a bounded subset of \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(V\) is a real Hilbert space compactly and densely embedded in \(L^p(\Omega)\) \((p\geq 2)\), \(A\) is a linear continuous and coercive operator, \(j\) is a locally Lipschitz function satisfying ``\(p-1\)-growth condition'' and \(j^0\) is the Clarke directional derivative.NEWLINENEWLINENEWLINEThe main result, stated for a general functional \(I\), is an alternative: either a number \(\overline\lambda\) satisfying some inequality is an eigenvalue of the problem \(\lambda Au\in\partial I(u)\) with \(u\in V\setminus\{0\}\), or we have a ``parametric representation'' of eigensolutions \((u,\lambda)\) of the problem NEWLINE\[NEWLINE\lambda Au\in\partial I(u),\quad \lambda> \overline\lambda,\tag{P}NEWLINE\]NEWLINE (in some particular sense). The proof is based on a nonsmooth version of the Mountain Pass Theorem (for its smooth version see [\textit{D. Motreanu}, Math. Slovaca 47, No. 4, 463-477 (1997; Zbl 0984.49026)]).NEWLINENEWLINENEWLINEAs an application of the main theorem, the author presents some results which describe properties of density and bifurcation for the eigensolutions of problem (P).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00041].