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

Abstract

Let Δ 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 − Δ k u ≥ | u | p in R N and the system of inequalities − Δ k u ≥ | v | p , − Δ 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.