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Forum Mathematicum

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Volume 30, Issue 5

Issues

Global integrability for solutions to some anisotropic problem with nonstandard growth

Hongya Gao
  • Corresponding author
  • College of Mathematics and Information Science, Hebei University, Baoding 071002, P. R. China
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/ Miaomiao Jia
Published Online: 2018-03-08 | DOI: https://doi.org/10.1515/forum-2017-0240

Abstract

This paper deals with the problem

u𝒦u*,ψ(Ω),v𝒦u*,ψ(Ω):Ωi=1n[ai(x,Du)-fi]Di(u-v)dxΩf(u-v)𝑑x,

where

{𝒦u*,ψ(Ω)={vu*+W01,(pi)(Ω):i=1nai(x,Du)DivL1(Ω) and vψ, a.e. Ω},u*W1,(pi)(Ω),θ=max{u*,ψ}u*+W01,(pi)(Ω),fL(p¯*)(Ω),fiLpi(Ω),i=1,,n,

and the Carathéodory functions ai:Ω×n, i=1,,n, satisfy some coercivity condition. We assume that the function θ=max{u*,ψ} makes ai(x,Dθ) to be more integrable than Lpi(Ω), i=1,,n, and then we prove that the solution u enjoys higher integrability.

Keywords: Global integrability; anisotropic problem; nonstandard growth

MSC 2010: 49N60; 35J60

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About the article


Received: 2017-11-10

Revised: 2017-12-03

Published Online: 2018-03-08

Published in Print: 2018-09-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11371050

Funding Source: Natural Science Foundation of Hebei Province

Award identifier / Grant number: A2015201149

The first author was supported by NSFC (grant no. 11371050) and NSF of Hebei Province (grant no. A2015201149).


Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1237–1243, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0240.

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