Semilinear equations associated with Dunkl Laplacian

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 $\mathbb R^d$.