Boundary value problems for elliptic equations of the form \(\sum_{| \alpha |,| \beta | \leq k}(-1)^{| \alpha |}D^{\alpha}(a_{\alpha \beta}D_ u^{\beta})=f\) on unbounded domains \(\Omega \subset {\mathbb{R}}^ n\) are studied. Using weighted Sobolev spaces, variants of the Poincaré and Friedrichs inequalities, and compact embeddings in weighted Sobolev spaces the author gives solvability conditions for the Dirichlet and Neumann problems.