The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases

In this paper, we investigate nonexistence results for stable solutions of the semilinear elliptic problem $$(-Δ)^s u+λ|x|^αu=|x|^β|u|^{p-1}u \; \mbox{in}\; \mathcal{H}, $$ where $\mathcal{H}=\mathbb{R}^n$ or $\mathcal{H}=\mathbb{R}^n_+=\{x=(x',x_n),\, x'\in\mathbb{R}^{n-1},x_n>0\}$ , with Dirichlet