Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains
Let \(\Omega=\omega\times \mathbb R^{N-m}\), where \(\omega\) is a bounded open set in \(\mathbb R^m\), \(m\geq 1\), \(N-m\geq 2\). Assume that \(F:\Omega\times \mathbb R\rightarrow \mathbb R\) is a Carathéodory function with subcritical growth. This paper is devoted to the study of the following eigenvalue problem for hemivariational inequalities: find \(u\in H^1_0(\Omega)\) such that \(\int_\Omega\nabla u\nabla w\,dx+\lambda\int_\Omega F^0(x,u(x);-w(x))\,dx\geq 0\) for all \(w\in H^1_0(\Omega)\), where \(F^0(x,s;t)\) denotes the Clarke generalized directional derivative of the locally Lipschitz mapping \(F(x,\cdot )\) at the point \(s\in \mathbb R\) in the direction \(t\in\mathbb R\). The first main result of the paper establishes the existence of at least three distinct, axially symmetric solutions for certain eigenvalues. Next, applying the fountain theorem, the author proves the existence of infinitely many axially symmetric solutions, but also of axially nonsymmetric solutions in certain dimensions. In both cases the principle of symmetric criticality plays a crucial role in proofs. The paper is carefully written and the reviewer believes that the methods developed in the paper can be applied to other classes of nonlinear eigenvalue problems.
scientific article; zbMATH DE number 2176209