PreprintCC0arXiv
On the classification of solutions to the Logarithmic Laplacian critical Choquard equation
Abstract
In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we classify the positive classical solutions, in a suitable integrability class, of the critical logarithmic Choquard problem ℒΔu = σu + 1/|u|22 (Gln(u)−4/N∫ℝNu2ln u dx)u in ℝN, where σ∈ℝ, N≥ 1, ℒΔ is the logarithmic Laplacian, and Gln(u)=ln(1/|x|4)*u2. We construct explicit solutions as limits of critical fractional Choquard bubbles and prove that every positive classical solution in the prescribed class has the form uσ,t(x) = eN/4(σ−BN,ℒ) BN,0(t/(t2+|x−x0|2))N/2, t>0, x0∈ℝN, where BN,0 =(Γ(N)/πN/2Γ(N/2))1/2, BN,ℒ =2ln 2+4ψ(N/2)−2ψ(N)+4/Nln BN,0.
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BibTeX
@misc{arora2026classification,
title = {{On the classification of solutions to the Logarithmic Laplacian critical Choquard equation}},
author = {Rakesh Arora and Jacques Giacomoni},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35669},
eprint = {2609.35669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35669},
note = {CC0},
}
RIS
TY - UNPB
TI - On the classification of solutions to the Logarithmic Laplacian critical Choquard equation
AU - Rakesh Arora
AU - Jacques Giacomoni
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35669
UR - https://arxiv.org/abs/2609.35669
AB - In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we classify the positive classical solutions, in a suitable integrability class, of the critical logarithmic Choquard problem ℒ_Δu = σu + 1/|u|_2^2 (G_ln(u)−4/N∫_ℝ^Nu^2ln u dx)u in ℝ^N, where σ∈ℝ, N≥ 1, ℒ_Δ is the logarithmic Laplacian, and G_ln(u)=ln(1/|x|^4)*u^2. We construct explicit solutions as limits of critical fractional Choquard bubbles and prove that every positive classical solution in the prescribed class has the form u_σ,t(x) = e^N/4(σ−B_N,ℒ) B_N,0(t/(t^2+|x−x_0|^2))^N/2, t>0, x_0∈ℝ^N, where B_N,0 =(Γ(N)/π^N/2Γ(N/2))^1/2, B_N,ℒ =2ln 2+4ψ(N/2)−2ψ(N)+4/Nln B_N,0.
ER -