The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases
Abstract
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−1u in ℋ, where ℋ=ℝn or ℋ=ℝn+={x=(x',xn), x'∈ℝn−1,xn>0} , with Dirichlet conditions in the half-space case. We assume that n≥2s, p>1, λ>0, α, β>−2s and 0<s≤1. Our aim is to analyze the influence of the Hardy-type power potential λ|x|αu in the presence of the Hénon weight |x|β, 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 ∞-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).
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BibTeX
@misc{fourti2026influence,
title = {{The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases}},
author = {Habib Fourti and Abdelbaki Selmi and Cherif Zaidi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05841},
eprint = {2610.05841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05841},
}
RIS
TY - UNPB
TI - The influence of a Generalized Hardy-Type Potential on Finite Morse Index Solutions of the Hénon Problem: Local and Nonlocal Cases
AU - Habib Fourti
AU - Abdelbaki Selmi
AU - Cherif Zaidi
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05841
UR - https://arxiv.org/abs/2610.05841
AB - 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−1u in ℋ, where ℋ=ℝ^n or ℋ=ℝ^n_+={x=(x',x_n), x'∈ℝ^n−1,x_n>0} , with Dirichlet conditions in the half-space case. We assume that n≥2s, p>1, λ>0, α, β>−2s and 0<s≤1. Our aim is to analyze the influence of the Hardy-type power potential λ|x|^αu in the presence of the Hénon weight |x|^β, 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 ^∞-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).
ER -