{"title":"Existence of very weak solutions to nonlinear elliptic equation with nonstandard growth and global weighted gradient estimates","authors":"Sun-Sig Byun , Minkyu Lim","doi":"10.1016/j.jde.2025.113594","DOIUrl":null,"url":null,"abstract":"<div><div>We study a general class of quasilinear elliptic equations with nonstandard growth to prove the existence of a very weak solution to such a problem. A key ingredient in the proof is a priori global weighted gradient estimate of a very weak solution, where the right-hand side of the equation is the divergence of a vector-valued function with a low degree of integrability. To obtain this estimate, we adopt a notion of reverse Hölder class of Muckenhoupt weights. Another crucial part of the proof is a generalized weighted div-curl lemma in the setting of Orlicz spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113594"},"PeriodicalIF":2.4000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625006217","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study a general class of quasilinear elliptic equations with nonstandard growth to prove the existence of a very weak solution to such a problem. A key ingredient in the proof is a priori global weighted gradient estimate of a very weak solution, where the right-hand side of the equation is the divergence of a vector-valued function with a low degree of integrability. To obtain this estimate, we adopt a notion of reverse Hölder class of Muckenhoupt weights. Another crucial part of the proof is a generalized weighted div-curl lemma in the setting of Orlicz spaces.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics