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 Holder 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 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)].

中文翻译：

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一类非标准生长条件非线性问题弱解的规律性

在本文中，我们研究了非线性问题 (1.2) 的弱解 u ∈ Vp(x) 的内部可微性，该问题出现在开放有界域 Ω⊂Rd, d = 2, 3 中的电流变流体 (ERF) 中。首先，通过建立反向 Holder 不等式，我们证明了 (1.2) 的弱解 u 具有满足 |Du|p(x)∈Llocδ(Ω) 且有一些 δ > 1 和 p(x)∈( 3dd+2,2)。接下来，基于Du的更高可积性，我们再通过差商理论和bootstrap论证推导出u的更高可微性，由此我们得到u的Holder连续性。这里，Diening 等人已经建立了（1.2）-（1.5）弱解的分析和存在理论。[具有可变指数的 Lebesgue 和 Sobolev 空间（Springer-Verlag Berlin Heidelberg，2011）]。