Equivalence between Superharmonic functions and renormalized solutions for the equations with $(p,q)$-growth

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.