# Caccioppoli-type inequalities for the Dunkl-$A$-Laplacian and their application to nonexistence result

## TL;DR

This paper develops Caccioppoli inequalities for the Dunkl-$A$-Laplacian operator and uses them to determine conditions under which certain solutions to Dunkl-differential inequalities cannot exist.

## Contribution

It introduces a new Dunkl-$A$-Laplacian operator and derives Caccioppoli inequalities within this framework, leading to nonexistence results for solutions.

## Key findings

- Derived local and global Caccioppoli inequalities for Dunkl-$A$-Laplacian.
- Established a sufficient condition for the nonexistence of solutions to Dunkl-differential inequalities.
- Extended classical inequalities to the Dunkl setting with applications to nonexistence proofs.

## Abstract

For a suitable function $A:\mathbb{R}^n\to \mathbb{R}^n$, we introduce the $A$-Laplacian in the Dunkl framework as $\Delta_{k,A}(u) =\text{div}_k(A(\nabla_ku))$, where $\nabla_k$ is the Dunkl-gradient operator associated with the multiplicity function $k$ and the root system $\mathcal{R}$. We derive the local and global Caccioppoli-type inequality for an element $u$ in the Dunkl-Orlicz-Sobolev space, satisfying the Dunkl-differential inequality $$ -\Delta_{k, A}(u) \geq b\Phi(u)\chi_{\{u>0\}}. $$ Using the Caccioppoli inequality, we establish a sufficient condition for the nonexistence of a nonzero solution $u$ to the Dunkl-differential inequality.