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

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Abstract

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).

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Acknowledgements

The author wishes to thank the anonymous reviewer for many valuable comments and suggestions to improve the expressions.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, China

    Fengping Yao

  2. Newtouch Center for Mathematics of Shanghai University, Shanghai, 200444, China

    Fengping Yao

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  1. Fengping Yao
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Correspondence to Fengping Yao.

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Communicated by Klaus Guerlebeck.

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Yao, F. Hölder continuity of the gradients for non-homogenous elliptic equations of p(x)-Laplacian type. Ann. Funct. Anal. 15, 40 (2024). https://doi.org/10.1007/s43034-024-00340-1

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