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Summability properties under nonstandard growth conditions

  • Ting Shan , Hongya Gao ORCID logo and Qianqian Liu EMAIL logo
Published/Copyright: October 11, 2025
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Abstract

This paper investigates summability properties under nonstandard growth conditions. We consider the boundary value problem of the 𝒜 -harmonic equation

{ - div 𝒜 ( x , u ( x ) ) = f ( x ) in  Ω , u ( x ) = u ( x ) on  Ω ,

where f ( x ) and D u * ( x ) lie in some Marcinkiewicz spaces. Summability properties of entropy solutions are obtained under nonstandard growth conditions on the operator 𝒜 : Ω × n n . We also consider T-minima of the integral functional

𝒥 ( v ) = Ω j ( x , v ( x ) ) 𝑑 x - Ω f ( x ) v ( x ) 𝑑 x ,

where f ( x ) lies in some Marcinkiewicz space. We consider T-minima among functions whose boundary values agree with u * ( x ) , and obtain some summability properties under nonstandard growth conditions of the operator j ( x , ξ ) : Ω × n .

MSC 2020: 35J25; 35J20

Award Identifier / Grant number: 12071021

Funding statement: The second author thanks the National Natural Science Foundation of China (Grant No. 12071021) and the Innovation Capacity Enhancement Program-Science and Technology Platform Project, Hebei Province (Grant No. 22567623H) for the support.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments.

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Received: 2024-10-31
Revised: 2025-04-25
Accepted: 2025-05-04
Published Online: 2025-10-11

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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