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

Abstract

In this paper, we study the interior differentiability of a weak solution u ∈ Vp(x) to a nonlinear problem (1.2), which arises in electroheological fluids (ERFs) in an open bounded domain Ω⊂Rd, d = 2, 3. At first, by establishing a reverse Hölder inequality, we show that the weak solution u of (1.2) has bounded energy that satisfies |Du|p(x)∈Llocδ(Ω) with some δ > 1 and p(x)∈(3dd+2,2). 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 Hölder 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)].