Let ∆ k be the Dunkl Laplacian relative to a fixed root system R in R d , d ≥ 2, and to a nonnegative multiplicity function k on R. Our first purpose in this paper is to solve the ∆ k-Dirichlet problem for annular regions. Secondly, we introduce and study the ∆ k-Green function of the annulus and we prove that it can be expressed by means of ∆ k-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for ∆ k-subharmonic functions and we study positive continuous solutions for a ∆ k-semilinear problem.