Hölder continuity of the gradients for non-homogenous elliptic equations of p(x)-Laplacian type

The main goal of this paper is to discuss the local Hölder continuity of the gradients for weak solutions of the following non-homogenous elliptic p(x)-Laplacian equations of divergence form

$$\begin{aligned} \text {div} \left( \left( A(x) \nabla u(x) \cdot \nabla u(x) \right) ^{\frac{p(x)-2}{2}} A(x) \nabla u(x) \right) = \text {div} \left( |\textbf{f}(x) |^{p(x)-2} \textbf{f}(x) \right) ~~ \text{ in }~ \Omega , \end{aligned}$$

where \(\Omega \subset \mathbb {R}^{n}\) is an open bounded domain for \(n \ge 2\), under some proper non-Hölder conditions on the variable exponents p(x) and the coefficients matrix A(x).