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

Abstract

For a suitable function $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$, we introduce the $A$-Laplacian in the Dunkl framework as ${\Delta }_{k,A}(u)={\text{div}}_k(A({\nabla }_{k}u))$, 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 $$-{\Delta }_{k,A}(u)\ge b\Phi (u){\chi }_{\{u>0\}}.$$ Using the Caccioppoli inequality, we establish a sufficient condition for the nonexistence of a nonzero solution $u$ to the Dunkl-differential inequality.

arXiv comment21 pages