Perturbations of even nonsmooth functionals
Let \(\Omega \subset \mathbb{R}^n\) \((n \geq 3)\) be a bounded domain and \(p : \Omega \times \mathbb{R} \to \mathbb{R}\) a Carathéodory function satisfying (1) \(p(x, - s) = - p(x,s)\), \(sp(x,s) \geq 0\), \(|p(x,s) |\leq a_1(x) + b |s |^r\), with \(a_1 \in L^{2n/(n + 2)} (\Omega)\), \(b \in \mathbb{R}\) and \(0 < r < (n + 2)/(n - 2)\). Assume that \(\psi : \Omega \to [0, + \infty]\) is a quasi-lower semicontinuous function; let \(\rho > 0\) be such that (2) \(\rho^2 < \int_\Omega \psi^2 dx\). Further, the constraint set \(\mathbb{K}\) is defined by \[ \mathbb{K} = \bigl\{ u \in H^1_0 (\Omega) : - \psi (x) \leq \widetilde u(x) \leq \psi (x) \text{ cap. q.e. in } \Omega \bigr\}, \tag{3} \] where \(\widetilde u\) is a quasi-continuous representative of \(u\). The authors study the nonlinear eigenvalue problem: \[ \int_\Omega \biggl[ DuD(v - u) + \bigl( p(x,u) + q(x,u) \bigr) (v - u) \biggr] dx + \langle \mu,v - u \rangle \geq \lambda \int_\Omega u(v - u) dx\;\forall v \in \mathbb{K}, \tag{4} \] \[ \int_\Omega u^2 dx = \rho^2,\;(\lambda,u) \in \mathbb{R} \times \mathbb{K} \] which represents a perturbed problem of the following one: \[ \bigl[ DuD(v - u) + p(x,u) (v - u) \bigr] dx \geq \lambda \int_\Omega u(v - u) dx\;\forall v \in\mathbb{K}, \tag{5} \] \[ \int_{\Omega} u^2 dx = \rho^2,\;(\lambda,u) \in \mathbb{R} \times \mathbb{K} \] Here \(q\) is a Carathéodory function and \(\mu \in H^{-1} (\Omega)\). The main result of the paper may now be formulated as follows. If \(\mathbb{K}\) and \(S_\rho = \{u : \int_\Omega u^2 dx = \rho^2\}\) are not tangent, then for every \(m \in \mathbb{N}\) there exists \(\delta > 0\) such that problem (4) has at least \(m\) solutions \((\lambda_1, u_1), \dots, (\lambda_m, u_m)\) with \(u_1, \dots, u_m\) all distinct. The paper reads well and constitutes a valuable contribution to eigenvalue problems arising in the theory of variational inequalities.
perturbations of variational inequalities
scientific article; zbMATH DE number 763451