Geodesic
A Compact Imbedding Theorem for Functions without Compact Support
Canadian mathematical bulletin, Tome 14 (1971) no. 3, pp. 305-309

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The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort 1 to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings 2 are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case. 
Adams, R. A.; Fournier, John. A Compact Imbedding Theorem for Functions without Compact Support. Canadian mathematical bulletin, Tome 14 (1971) no. 3, pp. 305-309. doi: 10.4153/CMB-1971-056-2
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     author = {Adams, R. A. and Fournier, John},
     title = {A {Compact} {Imbedding} {Theorem} for {Functions} without {Compact} {Support}},
     journal = {Canadian mathematical bulletin},
     pages = {305--309},
     year = {1971},
     volume = {14},
     number = {3},
     doi = {10.4153/CMB-1971-056-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1971-056-2/}
}
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