The aim of the paper is to prove the existence of solutions of an elliptic variational inequality involving a possibly unbounded domain \(\Omega \subset {\mathbb{R}}^ n\). To describe the considered problem more precisely let V be a closed subspace of the Sobolev space \(W^ m_ p(\Omega)\) and \(K\subset V\) a closed convex set containing 0. \(K^ 0\) denotes the set of all \(v\in K\) such that \(D^{\alpha}v\in L^{\infty}(\Omega)\) for \(| \alpha | \leq m\) and \(v(x)=0\) a.e. for sufficiently large \(| x|\). Furthermore, it is assumed that for each \(v\in K^ 0\) there are \(v_ j\in C^{\infty}_ 0({\mathbb{R}}^ n)\) such that \(v_{j|_{\Omega}}\in K\), \(v_{j|_{\Omega}}\to v\) strongly in \(W^ m_ p(\Omega)\) and \(\sup | D^{\alpha}v_ j| \leq c_{\alpha}.\) Under certain assumptions concerning the given functions \(f_{\alpha}\) and \(g_{\alpha}\) the author proves that for any linear continuous functional G over V there is a \(u\in K\), which solves the variational inequality \[ \begin{split} \sum_{|\alpha|\leq m} \int_{\Omega} f_{\alpha} (x,u,\ldots,D^{\beta}u,\ldots) (D^{\alpha}v-D^{\alpha}u) dx + \\ + \sum_{|\alpha|\leq m-1} \int_{\Omega} g_{\alpha} (x, u, \ldots, D^{\beta}u, \ldots) (D^{\beta}v - D^{\beta}u) dx \geq <G,v- u>, \\ \text{ for all }v\in K^ 0\quad (| \beta | \leq m). \end{split} \] It is proved that its solutions can be obtained as a limit of solutions of specified variational inequalities involving \(\Omega_ r = \Omega \cap B_ r\), \(B_ r = \left\{ x\in\mathbb{R}^ n \left| | x| <r\right.\right\}\).