On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R3, having the diffusion term Ap(u)=del(|D(u)|p-2D(u)) with D(u)=12(del u+(del u)inverted perpendicular), 3/2<p<3. In the case 3/2<p <= 9/5, we show that a suitable weak solution u is an element of W1,p(R3) satisfying lim infR ->infinity|uB(R)|=0 is trivial, i.e., u equivalent to 0. On the other hand, for 9/5<p<3 we prove the following Liouville type theorem: if there exists a matrix valued function V={Vij} such that partial derivative jVij=ui(summation convention), whose L3p2p-3 mean oscillation has the following growth condition at infinity, then u equivalent to 0.