The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth

TL;DR

This paper proves a weak Harnack inequality with optimal exponent for unbounded supersolutions of elliptic PDEs with generalized Orlicz growth, under suitable Lebesgue or Sobolev space conditions, and demonstrates the sharpness of these assumptions.

Contribution

It establishes the weak Harnack inequality for unbounded supersolutions in the context of generalized Orlicz growth, extending previous results with optimal conditions.

Findings

Weak Harnack inequality holds with optimal exponent.

Conditions on supersolutions are shown to be sharp.

Results apply to unbounded solutions in generalized Orlicz spaces.

Abstract

We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. We show that they satisfy the weak Harnack inequality with optimal exponent provided that they belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish the sharpness of our central assumptions.