Semilinear equations associated with Dunkl Laplacian

TL;DR

This paper investigates conditions for the existence of positive solutions to semilinear equations involving the Dunkl Laplacian, a differential operator linked to reflection groups, in bounded and unbounded domains.

Contribution

It establishes necessary and sufficient conditions for positive solutions of semilinear Dunkl Laplacian equations in the unit ball and entire space.

Findings

Derived criteria for solution existence in bounded domains.

Extended results to solutions in the whole space.

Connected Dunkl Laplacian properties with semilinear equations.

Abstract

Let $\Delta_k$ be the Dunkl Laplacian on $\mathbb R^d$ associated with a reflection group $W$ and a multiplicity function $k$. The purpose of this paper is to establish necessary and sufficient condition under which there exists a positive solution of the equation $\Delta_ku=\varphi(u)$ in the unit ball of $\mathbb R^d$ as well as in the whole space $\mathbb R^d$.