Liouville-type results for semilinear inequalities involving the Dunkl Laplacian operator

Abstract

Let Delta_k be the Dunkl generalized Laplacian operator associated to a root system R of R^N and a nonnegative multiplicity function k defined on R and invariant by the finite reflection group W. In this paper, we establish Liouville-type theorems for the semilinear inequality -Delta_k u >= |u|(p )in R-N and the system of inequalities -Delta_k u >= |v|(p), -Delta_k v >= |u|(q )in R^N, where N >= 1 and p, q >1. To the best of our knowledge, this contribution is the first work dealing with Liouville-type results for nonlinear problems involving the Dunkl Laplacian.