A fractional Brezis-Nirenberg problem with multipolar Hardy potentials
Abstract
We study a fractional Brezis--Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points a1,…,ak, with masses μ1,…,μk≥ 0. We assume ∑i=1kμi<ΛN,s and consider N>4s and 0<μmax<μ̅N,s. The interaction between the poles allows existence for λ=0 and for some negative values of λ. At a pole aj, this interaction is described by Bj(λ,a,μ)=λ+∑i≠ jμi/(|ai−aj|2s). This leads to a threshold λgeo≤ 0. We prove that the critical Rayleigh quotient is attained for λgeo<λ<λ1(μ,a), and hence the problem admits a positive least-energy solution. If at least two masses are positive, then λgeo<0, so the result includes λ=0 and a nonempty interval of negative values of λ. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.
Related
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BibTeX
@misc{mukherjee2026fractional,
title = {{A fractional Brezis-Nirenberg problem with multipolar Hardy potentials}},
author = {Debangana Mukherjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22877},
eprint = {2609.22877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22877},
}
RIS
TY - UNPB
TI - A fractional Brezis-Nirenberg problem with multipolar Hardy potentials
AU - Debangana Mukherjee
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22877
UR - https://arxiv.org/abs/2609.22877
AB - We study a fractional Brezis--Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points a_1,…,a_k, with masses μ_1,…,μ_k≥ 0. We assume ∑_i=1^kμ_i<Λ_N,s and consider N>4s and 0<μ_max<μ̅_N,s. The interaction between the poles allows existence for λ=0 and for some negative values of λ. At a pole a_j, this interaction is described by B_j(λ,a,μ)=λ+∑_i≠ jμ_i/(|a_i−a_j|^2s). This leads to a threshold λ_geo≤ 0. We prove that the critical Rayleigh quotient is attained for λ_geo<λ<λ_1(μ,a), and hence the problem admits a positive least-energy solution. If at least two masses are positive, then λ_geo<0, so the result includes λ=0 and a nonempty interval of negative values of λ. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.
ER -