Compact imbeddings between variable exponent spaces with unbounded underlying domain

In this paper, we have established some compact imbedding theorems for some subspaces of W^1,p(x)(U)$W^{1,p\left(x\right)}\left(U\right)$ when the underlying domain U$U$ is unbounded. The domain we consider is mainly of type R^N(N≥2)$R^N(N\ge 2)$ or R^L×Ω(L≥2)$R^L\times \Omega (L\ge 2)$, where Ω⊂R^M$\Omega \subset R^M$ is a bounded domain with smooth boundary.