Compact Imbedding Theorems For Variable Exponent Spaces With Unbounded Domains And Their Applications

In this paper,we have established some compact imbedding theorems for some subspaces of W^1,p(x)(U)when the domain U is unbounded.The domain we consider is mainly of type R^N(N≥2)or R^L×Ω(L≥2),whereΩ(?)R^M is a bounded domain with smooth boundary.Besides,taking advantage of our new-established compact imbedding theorems,we have investigated the existence of weak solutions to p(x)-Laplacian equations in the whole space R^N.In this paper,we have mainly focused ourselves on the Mountain-pass solutions of p(x)-Laplacian equations with prescribed symmetric and asymmetric properties.The p(x)-Laplacian equations with concave and convex nonlinearities are also considered.Under the basis of our paper,a great number of existence results on the weak solutions to p(x)-Laplacian equations in the whole space R^N could be obtained.