Existence of very weak solutions to nonlinear elliptic equation with nonstandard growth and global weighted gradient estimates

2023/11/20 by Sun‐Sig Byun, Byun, Sun-Sig, Minkyu Lim +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2311.11479

openalex publication_date 2023/11/20 · openalex created_date 2023/11/23 · openalex updated_date 2026/07/28

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 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.

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