Compact imbeddings between variable exponent spaces with unbounded underlying domain

In this paper, we have established some compact imbedding theorems for some subspaces of W1,p(x)(U)W1,p(x)(U) when the underlying domain UU is unbounded. The domain we consider is mainly of type RN(N≥2)RN(N≥2) or RL×Ω(L≥2)RL×Ω(L≥2), where Ω⊂RMΩ⊂RM is a bounded domain with smooth boundary.