On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball

Abstract

Abstract Let Δ k {\Delta }_{k} be the Dunkl generalized Laplacian operator associated with a root system R R of R N {{\mathbb{R}}}^{N} , N ≥ 2 N\ge 2 , and a nonnegative multiplicity function k k defined on R R and invariant by the finite reflection group W W . In this study, we study the existence and nonexistence of weak solutions to the semilinear inequality − Δ k u + λ ∣ x ∣ 2 u ≥ ∣ u ∣ p -{\Delta }_{k}u+\frac{\lambda }{{| x| }^{2}}u\ge {| u| }^{p} in R N \ B 1 ¯ {{\mathbb{R}}}^{N}\backslash \overline{{B}_{1}} under the boundary condition u ≥ 0 u\ge 0 on ∂ B 1 \partial {B}_{1} , where p > 1 p\gt 1 , λ ≥ − ( N − 2 + 2 γ ) 2 ⁄ 4 \lambda \ge -{(N-2+2\gamma )}^{2}/4 , and B 1 {B}_{1} is the open unit ball of R N {{\mathbb{R}}}^{N} . Namely, we show that the dividing line with respect to existence and nonexistence is given by a critical exponent that depends on λ \lambda , N N , and γ ( k ) \gamma \left(k) , where γ ( k ) = ∑ α ∈ R + k ( α ) \gamma \left(k)={\sum }_{\alpha \in {R}^{+}}k\left(\alpha ) and R + {R}^{+} is the positive subsystem.