Regularity of weak solutions to a class of nonlinear problem with non-standard growth conditions

In this paper, we study the interior differentiability of a weak solution u is an element of V-p(x) to a nonlinear problem (1.2), which arises in electroheological fluids (ERFs) in an open bounded domain Omega subset of Rd, d = 2, 3. At first, by establishing a reverse Holder inequality, we show that the weak solution u of (1.2) has bounded energy that satisfies |Du|p(x)is an element of Lloc delta(Omega) with some delta > 1 and p(x)is an element of(<mml:mfrac>3dd+2</mml:mfrac>,2). Next, based on the higher integrability of Du, we then derive the higher differentiability of u by the theory of difference quotient and a bootstrap argument, from which we obtain the Holder continuity of u. Here, the analysis and the existence theory of the weak solution to (1.2)-(1.5) have been established by Diening et al. [Lebesgue and Sobolev Spaces with Variable Exponents (Springer-Verlag Berlin Heidelberg, 2011)].