Semilinear equations associated with Dunkl Laplacian

Let ${\mathrm{\Delta }}_k$ 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 ${\mathrm{\Delta }}_{k}u=\phi (u)$ in the unit ball of ${\mathbb{R}}^d$ as well as in the whole space ${\mathbb{R}}^d$.