Kufner, A.; Opic, B.; Skrypnik, I. V.; Stetsyuk, V. P. [Opits, B.] Exact imbedding theorems for a certain class of weighted Sobolev spaces. (Russian. English summary) Zbl 0636.46034 Dokl. Akad. Nauk Ukr. SSR, Ser. A 1988, No. 1, 22-26 (1988). Let \(\Omega\subset {\mathbb{R}}^ N \)be a bounded domain, let L q(\(\Omega\) ;w) and \(W^{1,p}(\Omega;v_ 0,v_ 1)\) be normed by \(\| u\|_{q,w}=(\int_{\Omega}| u(x)|\) q w(x)dx)\({}^{1/q}\) and \((\| u\|\) \(p_{p,v_ 0}+\sum^{N}_{k=1}\| \partial u/\partial x_ k\|\) \(p_{p,v_ 1})^{1/p}\), respectively, \(1\leq q<\infty\), \(1\leq p<\infty\), and w, \(v_ 0\), \(v_ 1\) are positive weight functions. Of special interest is the model case \(\Omega =\{x|\) \(0<| x-x_ 0| <R\}\) where the weight functions depend only on \(| x-x_ 0|\). Let \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)\) be the completion of \(C^{\infty}_ 0(\Omega)\) in \(W^{1,p}(\Omega;v_ 0,v_ 1)\). The paper deals with embeddings and compact embeddings of the type \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)L\) q(\(\Omega\) ;w) and \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)L\) q(\(\Omega\) ;w), respectively. The conditions have necessary and sufficient character. Examples are studied: power weights, weights with exponential growth. Reviewer: H.Triebel MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:Sobolev spaces; compact embeddings; power weights; weights with exponential growth PDFBibTeX XMLCite \textit{A. Kufner} et al., Dokl. Akad. Nauk Ukr. SSR, Ser. A 1988, No. 1, 22--26 (1988; Zbl 0636.46034)