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 $$(-\mathrm{\Delta }{)}^{s}u+\lambda \mathrm{\mid }x{\mathrm{\mid }}^{\alpha }u=\mathrm{\mid }x{\mathrm{\mid }}^{\beta }\mathrm{\mid }u{\mathrm{\mid }}^{p-1}uin\mathcal{H},$$ where $\mathcal{H}={\mathbb{R}}^n$ or $\mathcal{H}={\mathbb{R}}_{+}^n=\{x=(x^{\mathrm{\prime }},x_n),x^{\mathrm{\prime }}\in {\mathbb{R}}^{n-1},x_n>0\}$ , with Dirichlet conditions in the half-space case. We assume that $n\ge 2s$, $p>1$, $\lambda >0,$ $\alpha ,\beta >-2s$ and $0<s\le 1$. Our aim is to analyze the influence of the Hardy-type power potential $\lambda \mathrm{\mid }x{\mathrm{\mid }}^{\alpha }u$ in the presence of the Hénon weight $\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$, with particular emphasis on the existence and classification of weak stable solutions, including solutions that may be unbounded or sign-changing. We establish nonexistence results for nontrivial stable solutions and, more generally, for solutions that are stable outside a compact set, covering critical and supercritical regimes without additional assumptions, and the subcritical regime under a suitable $L^{\mathrm{\infty }}$-smallness condition on a weighted version of the solution. Our results apply to both the local case $s=1$ and the nonlocal case $0<s<1$, in the whole space and in the half-space. The proofs are based on energy estimates combined with a Pohozaev identity. In the supercritical regime, we further establish a monotonicity formula, following the approach of Dávila et al. (Trans. Amer. Math. Soc. 369, 6087-6104, 2017).