Variational theorems of mixed type and asymptotically linear variational inequalities
The following variational inequality \[ \int_{\Omega} (DuD(v - u) - g(x,u)(v - u) + h(v - u))dx \geq 0 \quad \forall v \in K_{\Psi}, \] \[ u \in K_{\Psi} = \{ u \in W_{0}^{1,2} (\Omega); \;u \geq \Psi \text{ a.e.}\} \] where \(\Omega \subset \mathbb{R}^n \) is a bounded open subset with smooth boundary, \( g: \Omega \times\mathbb{R} \mapsto\mathbb{R} \) a function satisfying the Carathéodory conditions, \( \Psi: \Omega \mapsto (-\infty,\infty \left. \right] \) is a measurable function and \( h \) is a given function in \( L^2(\Omega) \), is considered. The authors estimate the number of solutions to the mentioned variational inequality in dependence on the parameter \( \alpha \) which appears in the condition \[ \lim_{s \to +\infty} {g(x,s)\over{s}} = \alpha\quad \text{ for a.e. } x \in \Omega. \] To obtain the results the authors apply a different technique than that in [\textit{A. Marino} and \textit{D. Passaseo}, Pitman Res. Notes Math. Ser. 243, 127-143 (1992; Zbl 0818.35034)] used for the linear case of \( g \). This new technique has been recently introduced in [\textit{A. Marino} and \textit{C. Saccon}, Ann. Sc. Norm. Super. Pisa, CI. Sci. (4) (to appear)].
scientific article; zbMATH DE number 1279188
scientific article