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Unilateral problems with quadratic growth in unbounded domains. (English) Zbl 0602.49008
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 $$\Omega$$.
Reviewer: H.Böttger

##### MSC:
 49J40 Variational inequalities 35D05 Existence of generalized solutions of PDE (MSC2000) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35J60 Nonlinear elliptic equations