Small weights in Caccioppoli's inequality and applications to Liouville-type theorems for non-standard problems

TL;DR

This paper introduces a modified Caccioppoli inequality with small weights to prove Liouville-type theorems for problems with non-standard growth conditions, expanding the scope of classical results.

Contribution

It develops a new variant of Caccioppoli's inequality involving small weights, enabling Liouville theorems under broader non-standard growth assumptions.

Findings

Established Liouville-type theorems for non-standard growth problems

Introduced a weighted Caccioppoli inequality with small weights

Extended classical results to more general settings

Abstract

Using a variant of Caccioppoli's inequality involving small weights, i.e. weights of the form $(1+|\nabla u|^2)^{-\alpha/2}$ for some $\alpha > 0$, we establish several Liouville-type theorems under general non-standard growth conditions.