Sharp concentration levels and extremal functions for a planar Choquard-type Trudinger--Moser inequality
Abstract
Let Ω⊂ℝ2 be a smooth bounded domain and let 0<α<2. We study the exponential Choquard functional ℱβ(u)=∫Ω∫Ω eβu2(x)eβu2(y)/(|x−y|α)d xd y on the unit Dirichlet ball of H01(Ω). We establish the sharp boundedness threshold β*=(4−α)π and prove attainment of the critical supremum. More precisely, we determine the exact concentration level at every point of Ω̅ in terms of the regular part of the Dirichlet Green function. The concentration upper bound follows from the classical planar Trudinger--Moser concentration formula and the sharp Hardy--Littlewood--Sobolev inequality, after separating the background contribution. A test function obtained by matching a logarithmic profile to the Green function attains this bound asymptotically. Its exact Dirichlet normalization produces a strictly positive lower-order correction, which rules out concentration of maximizing sequences. We also prove the Euler--Lagrange equation, positivity, and regularity of extremals.
Related
- Extremal Trudinger-Moser inequality associated with many point vortices with multi-intensities
- Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ℤN
- Trudinger-Moser type inequality in fractional Sobolev space with singularity on smooth submanifold
- The Trudinger inequality is true for M1,s spaces
- A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
BibTeX
@misc{gao2026sharp,
title = {{Sharp concentration levels and extremal functions for a planar Choquard-type Trudinger--Moser inequality}},
author = {Jinkai Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25313},
eprint = {2610.25313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25313},
}
RIS
TY - UNPB
TI - Sharp concentration levels and extremal functions for a planar Choquard-type Trudinger--Moser inequality
AU - Jinkai Gao
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25313
UR - https://arxiv.org/abs/2610.25313
AB - Let Ω⊂ℝ^2 be a smooth bounded domain and let 0<α<2. We study the exponential Choquard functional ℱ_β(u)=∫_Ω∫_Ω e^βu^2(x)e^βu^2(y)/(|x−y|^α)d xd y on the unit Dirichlet ball of H_0^1(Ω). We establish the sharp boundedness threshold β_*=(4−α)π and prove attainment of the critical supremum. More precisely, we determine the exact concentration level at every point of Ω̅ in terms of the regular part of the Dirichlet Green function. The concentration upper bound follows from the classical planar Trudinger--Moser concentration formula and the sharp Hardy--Littlewood--Sobolev inequality, after separating the background contribution. A test function obtained by matching a logarithmic profile to the Green function attains this bound asymptotically. Its exact Dirichlet normalization produces a strictly positive lower-order correction, which rules out concentration of maximizing sequences. We also prove the Euler--Lagrange equation, positivity, and regularity of extremals.
ER -