Equivalence between Superharmonic functions and renormalized solutions for the equations with $(p, q)$-growth
Ying Li, Chao Zhang
TL;DR
The paper addresses the problem of linking two solution concepts for elliptic equations with nonstandard $(p,q)$-growth and measure data: local renormalized solutions and $q$-superharmonic functions. Using Wolff potential estimates and a detailed measure decomposition, it proves that every local renormalized solution has a $q$-superharmonic representative and that every finite $q$-superharmonic function with its Riesz measure is a local renormalized solution, establishing a full equivalence in the $(p,q)$-growth setting. This extends classical p-growth results to nonstandard growth and provides a robust potential-theoretic framework for representing measure-data solutions. The work advances understanding of existence, regularity, and representation of solutions to nonlinear elliptic problems with measure data under growth conditions between two power laws.
Abstract
We establish the equivalence between superharmonic functions and locally renormalized solutions for the elliptic measure data problems with $(p, q)$-growth. By showing that locally renormalized solutions are essentially bounded below and using Wolff potential estimates, we extend the results of [T. Kilpeläinen, T. Kuusi, A. Tuhola-Kujanpää, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2011] to a broader class of problems. Our work provides the first equivalence result between locally renormalized solutions and superharmonic functions for the nonstandard growth equations.