Compact imbedding of weighted Sobolev space defined on an unbounded domain. I. (English) Zbl 0646.46029

Opic, Bohumír

Čas. Pěstování Mat. 113, No. 1, 60-73 (1988).

Let \(\Omega\) be an unbounded domain in \({\mathbb{R}}^ N,\) then \(W^{k,p}(\Omega;S)\) with \(1\leq p<\infty\) and \(k\in {\mathbb{N}}\) is the collection of distributions u such that \[ (\sum_{| \alpha | \leq k}\int_{\Omega}| D^{\alpha} u(x)| \quad p w_{\alpha}(x)dx)^{1/p}< \infty, \] where \(S=\{w_{\alpha}\}\) stands for the weight functions. \(W_ 0^{k,p}(\Omega;S)\) denotes the completion of \(C^{\infty}_ 0(\Omega)\) in \(W^{k,p}(\Omega;S)\). The aim of the paper is to derive conditions on S and on the weight function \(\rho\) such that the natural embedding from \(W_ 0^{k,p}(\Omega;S)\) into \(L^ p(\Omega;\rho)\) is compact.

Reviewer: H.Triebel

MSC:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

weighted Sobolev space defined on an unbounded domain; embedding