The authors prove the following existence theorem for a variational inequality with obstacle: Let \(\Omega \subset {\mathbb{R}}^ n\) (n\(\geq 1)\) be an unbounded open set with not necessarily smooth boundary and \(\psi\) : \(\Omega\to {\mathbb{R}}\) be a measurable function such that for \(K(\psi):=\{v\in H^ 1_ 0(\Omega)\), \(v\geq \psi\) a.e. in \(\Omega\}\) we have \(K(\psi)\cap L^{\infty}(\Omega)\neq \emptyset\); then there exists at least one \(u\in K(\psi)\cap L^{\infty}(\Omega)\) satisfying \[ \int_{\Omega}\{[\sum^{n}_{i,j=1}(\partial (v-u)/\partial x_ i)a_{ij}(x,u)\partial u/\partial x_ j]+[ua_ 0(x)+f(x,u,\nabla u)](u-v)\}dx\geq 0 \] for all \(v\in K(\psi)\cap L^{\infty}(\Omega)\), where the coefficients \(a_{ij}: \Omega \times {\mathbb{R}}\to {\mathbb{R}}\) and the nonlinear term \(f: \Omega\times {\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}\) are Carathéodory functions and \[ \sum_{i,j}a_{ij}(x,s)\xi_ i\xi_ j\geq \alpha | \xi |^ 2\quad (\alpha >0) \] for \(\xi =(\xi_ i)\in {\mathbb{R}}^ n\), \(0<\alpha_ 0\leq a_ 0(x)\leq \beta_ 0\) for almost all \(x\in \Omega\) and all \(s\in {\mathbb{R}}| f(x,s,\xi)| \leq \rho (x)+b(| s|)[h(x)| \xi | +| \xi |^ 2]\)a.e. in \(\Omega\) for all \(s\in {\mathbb{R}}\), \(\xi \in {\mathbb{R}}^ n\) with an increasing function \(b: {\mathbb{R}}^+\to {\mathbb{R}}^+\) and \(\rho \in L^ 2(\Omega)\cap L^{\infty}(\Omega)\), \(h\in L^ p(\Omega)\cap L^{\infty}(\Omega)\) for some \(p\in [1,\infty).\)
This extends a result by Boccardo, Murat and Puel who proved virtually the same existence theorem under the assumption that \(\Omega \subset {\mathbb{R}}^ n\) is a bounded open set [cf. L. Boccardo, F. Murat and J. P. Puel, Nonlinear partial differential equations and their applications, Coll. France Semin., Vol. 4, Res. Notes Math. 84, 19- 73 (1983; Zbl 0588.35041)].
In the paper under review a modification of the proof by Boccardo-Murat- Puel is given which works in the case of an unbounded domain \(\Omega\).