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

Abstract

Let Δk{\Delta }_{k} be the Dunkl generalized Laplacian operator associated with a root system RR of RN{{\mathbb{R}}}^{N}, N≥2N\ge 2, and a nonnegative multiplicity function kk defined on RR and invariant by the finite reflection group WW. In this study, we study the existence and nonexistence of weak solutions to the semilinear inequality −Δku+λ∣x∣2u≥∣u∣p-{\Delta }_{k}u+\frac{\lambda }{{| x| }^{2}}u\ge {| u| }^{p} in RN\B1¯{{\mathbb{R}}}^{N}\backslash \overline{{B}_{1}} under the boundary condition u≥0u\ge 0 on ∂B1\partial {B}_{1}, where p>1p\gt 1, λ≥−(N−2+2γ)2⁄4\lambda \ge -{(N-2+2\gamma )}^{2}/4, and B1{B}_{1} is the open unit ball of RN{{\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 , NN, 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.

Keywords

• semilinear inequalities
• dunkl operators
• inverse-square potential
• existence
• nonexistence
• critical exponent
• 35r45
• 35a01
• 35b33
• 35j91