On a Semilinear Inequality with Dunkl Laplacian and Inverse-Square Potential on a Ball

Let \(R\subset \mathbb {R}^N\backslash \{0\}\) $R\subset {\mathbb{R}}^N\backslash \{0\}$ be a root system and \(R^+\) $R^{+}$ be a positive subsystem. Let k be a nonnegative multiplicity function defined on R and invariant by the the reflection group and \(\gamma =\sum _{\alpha \in R^+}k(\alpha )\) $\gamma ={\sum }_{\alpha \in R^{+}}k(\alpha )$ . This paper is devoted to study the semilinear inequality \(\begin{aligned} -\Delta _ku\ge \frac{\lambda }{|x|^2} u+|x|^a |u|^p \text{ in } B_1\backslash \{0\},\quad u\ge 0 \text{ on } \partial B_1, \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{k}u\ge \frac{\lambda }{{|x|}^2}u+{|x|}^a{|u|}^p\text{in}{B}_1\backslash \{0\},u\ge 0\text{on}\partial B_1,\end{array}$$ where \(\Delta _k\) ${\mathrm{\Delta }}_k$ is the Dunkl Laplacian operator associated to R and k, \(B_1\) $B_1$ is the open unit ball of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ , \(N\ge 2\) $N\ge 2$ , \(p>1\) $p>1$ , \(a\in \mathbb {R}\) $a\in \mathbb{R}$ and \(\lambda <\left( \frac{N-2+2\gamma }{2}\right) ^2\) $\lambda <{\left(\frac{N-2+2\gamma }{2}\right)}^2$ . More precisely, we first establish a general sharp nonexistence result for the considered problem. Then, we discuss separately the cases \(\lambda <0\) $\lambda <0$ , \(\lambda =0\) $\lambda =0$ and \(0<\lambda <\left( \frac{N-2+2\gamma }{2}\right) ^2\) $0<\lambda <{\left(\frac{N-2+2\gamma }{2}\right)}^2$ . To the best of our knowledge, the issue of existence and nonexistence of solutions to nonlinear problems involving Dunkl operators has not been previously studied in the literature.