Sobolev embeddings for unbounded domain with variable exponent having values across N
Xianling Fan · Mathematical Inequalities & Applications · 2010
We study Sobolev embeddings for unbounded domain with variable exponent having values across N . A main result of this paper is the following theorem: Let be an unbounded domain in R N satisfying the uniform cone condition. Suppose that p : R is Lipschitz and 1 < inf p sup p < . Then there holds a continuous embedding W 1,p() () L q() () for any q L () satisfying condition p(x) q(x) p * (x) for a.e. x , where p * (x) = Np(x) N-p(x) if p(x) < N and p * (x) = if p(x) N . In this theorem the usual condition sup p(x) < N is not required.