Holder continuity of weak solution to a nonlinear problem with non-standard growth conditions

We study the Holder continuity of weak solution u to an equation arising in the stationary motion of electrorheological fluids. To this end, we first obtain higher integrability of Du in our case by establishing a reverse Holder inequality. Next, based on the result of locally higher integrability of Du and difference quotient argument, we derive a Nikolskii type inequality; then in view of the fractional Sobolev embedding theorem and a bootstrap argument we obtain the main result. Here, the analysis and the existence theory of a weak solution to our equation are similar to the weak solution which has been established in the literature with 3d/d+2 <= p(infinity) <= p(x) <= p(0) < infinity, and in this paper we confine ourselves to considering p(X) is an element of (3d/d+2, 2) and space dimension d = 2, 3.