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. Kilpel\"{a}inen, T. Kuusi, A. Tuhola-Kujanp\"{a}\"{a}, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 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.