Profile decomposition, strict binding and global compactness for a critical Grushin embedding
Abstract
We study attainment of the best constant in ∫ℝm+n (|∇xu|2+|x|2|∇yu|2+a|u|2) ≥ Sa(∫ℝm+n (1+|x|2)n/(Q−2)|u|q)2/q, where m,n≥1, Q=m+2n, q=2Q/(Q−2), and a>0. Using annular interpolation and a transverse-average recurrence with Lq contraction 2−m/q, we decompose bounded sequences into finite, Grushin and Euclidean profiles. Energy and critical mass split, with a power bound in the number of profiles on the remaining mass. We prove compactness of normalized minimizing sequences modulo y-translations under Sa<min{SG,an/QSE(1)}, where SG and SE(1) are the tangent constants. The strict-binding set is empty or an open interval; we prove nonemptiness for n=1 and even m≥6. At finite positive endpoints of a nonempty binding interval, we characterize attainment by first-order binding gaps. We obtain a finite critical-profile decomposition of Palais--Smale sequences with a remainder vanishing in energy.
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BibTeX
@misc{deng2026profile,
title = {{Profile decomposition, strict binding and global compactness for a critical Grushin embedding}},
author = {Shengbing Deng and Dehong Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01094},
eprint = {2611.01094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01094},
note = {CC0},
}
RIS
TY - UNPB
TI - Profile decomposition, strict binding and global compactness for a critical Grushin embedding
AU - Shengbing Deng
AU - Dehong Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01094
UR - https://arxiv.org/abs/2611.01094
AB - We study attainment of the best constant in ∫_ℝ^m+n (|∇_xu|^2+|x|^2|∇_yu|^2+a|u|^2) ≥ S_a(∫_ℝ^m+n (1+|x|^2)^n/(Q−2)|u|^q)^2/q, where m,n≥1, Q=m+2n, q=2Q/(Q−2), and a>0. Using annular interpolation and a transverse-average recurrence with L^q contraction 2^−m/q, we decompose bounded sequences into finite, Grushin and Euclidean profiles. Energy and critical mass split, with a power bound in the number of profiles on the remaining mass. We prove compactness of normalized minimizing sequences modulo y-translations under S_a<min{S_G,a^n/QS_E(1)}, where S_G and S_E(1) are the tangent constants. The strict-binding set is empty or an open interval; we prove nonemptiness for n=1 and even m≥6. At finite positive endpoints of a nonempty binding interval, we characterize attainment by first-order binding gaps. We obtain a finite critical-profile decomposition of Palais--Smale sequences with a remainder vanishing in energy.
ER -