Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions
Abstract
We study the Brezis--Nirenberg problem for the fractional Laplacian with an upper critical Hartree nonlinearity on a bounded domain Ω⊂ℝN, in the low-dimensional range 2s<N<4s. For a continuous potential a, we prove that the following properties are equivalent: the Robin function of (−Δ)s+a is negative somewhere in Ω; the associated Hartree--Sobolev quotient 𝒮H,s(a) lies strictly below the best constant 𝒮H,s; and 𝒮H,s(a) is attained. This leads to the notion of a critical potential, for which 𝒮H,s(a)=𝒮H,s and every nontrivial decrease of a lowers the infimum. For a critical potential a and V∈ L∞(Ω), we determine the leading term in the asymptotics of 𝒮H,s(a+ε V) as ε→0+ and describe the blow-up of almost-minimizers, including the concentration speed and the location of the concentration point. Compared with the Sobolev case, the nonlocal nonlinearity produces several new interaction terms; a key ingredient is a proof of non-attainment at the threshold based on the sharp Hardy--Littlewood--Sobolev inequality on the sphere.
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BibTeX
@misc{vaira2026critical,
title = {{Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions}},
author = {Giusi Vaira and Yue Wu and Minbo Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26161},
eprint = {2610.26161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26161},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical functions and asymptotics for fractional Hartree Brezis--Nirenberg problems in low dimensions
AU - Giusi Vaira
AU - Yue Wu
AU - Minbo Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26161
UR - https://arxiv.org/abs/2610.26161
AB - We study the Brezis--Nirenberg problem for the fractional Laplacian with an upper critical Hartree nonlinearity on a bounded domain Ω⊂ℝ^N, in the low-dimensional range 2s<N<4s. For a continuous potential a, we prove that the following properties are equivalent: the Robin function of (−Δ)^s+a is negative somewhere in Ω; the associated Hartree--Sobolev quotient 𝒮_H,s(a) lies strictly below the best constant 𝒮_H,s; and 𝒮_H,s(a) is attained. This leads to the notion of a critical potential, for which 𝒮_H,s(a)=𝒮_H,s and every nontrivial decrease of a lowers the infimum. For a critical potential a and V∈ L^∞(Ω), we determine the leading term in the asymptotics of 𝒮_H,s(a+ε V) as ε→0^+ and describe the blow-up of almost-minimizers, including the concentration speed and the location of the concentration point. Compared with the Sobolev case, the nonlocal nonlinearity produces several new interaction terms; a key ingredient is a proof of non-attainment at the threshold based on the sharp Hardy--Littlewood--Sobolev inequality on the sphere.
ER -