Regularity for very weak solutions to elliptic equations of $ p $-Laplacian type

Qing Zhao; Shuhong Chen; Qing Zhao; Shuhong Chen

doi:10.3934/era.2025154

Electronic Research Archive

2025, Volume 33, Issue 6: 3482-3495. doi: 10.3934/era.2025154

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Research article

Regularity for very weak solutions to elliptic equations of $ p $-Laplacian type

Qing Zhao ¹,
Shuhong Chen ^{2,3
,
,}

1.
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
2.
School of Information Engineering, Zhejiang Ocean University, Zhoushan 316022, China
3.
School of Mathematics and Computer, Wuyi University, Wuyishan 354300, China

Received: 10 March 2025 Revised: 28 April 2025 Accepted: 21 May 2025 Published: 05 June 2025

We study the regularity problem with non-homogeneous terms of $ p $-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.
- regularity theory,
- generalized weak solutions,
- Hodge-type decomposition,
- elliptic equations,
- $ p $-Laplacian class
Citation: Qing Zhao, Shuhong Chen. Regularity for very weak solutions to elliptic equations of $ p $-Laplacian type[J]. Electronic Research Archive, 2025, 33(6): 3482-3495. doi: 10.3934/era.2025154

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Abstract

We study the regularity problem with non-homogeneous terms of $ p $-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.

References

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