Embedding Theorems and Quasi-Linear Elliptic Boundary Value Problems for Unbounded Domains

The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in <!-- MATH ${{\mathbf{R}}^N}$ --> .