Existence and non-existence of extremal functions for inequalities of Bliss-Moser type
Abstract
We establish a concentration--compactness theory for a one-dimensional singular Moser-type functional with logarithmic weight function supv∈ E ∫01 e(loge/s+γlogloge/s)v2(s)/sds<∞, γ∈[0,1], where E:= {u ∈ H1(0, 1): u(0) = 0, ∫01 |u'|2dx=1}. In particular, we prove compactness for all normalized sequences in the subcritical range γ<1, thereby resolving an open problem proposed in [J. M. do Ó, B. Ruf and P. Ubilla, CVPDE 2023]. We also study the critical Bliss-Moser inequality: nonumber supv∈ E Ic(v)< +∞, Ic(v)=∫01 e(loge/s+logloge/s+cs)v2(s)/sds. For any c∈(−∞,+∞), we determine the sharp concentration level supvj∈ E,vj is NCSlimsupj→∞ Ic(vj)=1+e2, and obtain explicit sufficient conditions for attainability under c>c* and non-attainability under c<−c*. This results describe how linear perturbations govern the transition between compactness, concentration, and attainability at the critical level.
Related
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- Extremal Trudinger-Moser inequality associated with many point vortices with multi-intensities
- A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
- The compactness of Moser-Trudinger type inequalities in the unit ball
- Extremals for Trudinger-Moser inequalities via polygonal profiles
BibTeX
@misc{guo2026existence,
title = {{Existence and non-existence of extremal functions for inequalities of Bliss-Moser type}},
author = {Yanyan Guo and Huxiao Luo and Bernhard Ruf},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23251},
eprint = {2609.23251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23251},
}
RIS
TY - UNPB
TI - Existence and non-existence of extremal functions for inequalities of Bliss-Moser type
AU - Yanyan Guo
AU - Huxiao Luo
AU - Bernhard Ruf
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.23251
UR - https://arxiv.org/abs/2609.23251
AB - We establish a concentration--compactness theory for a one-dimensional singular Moser-type functional with logarithmic weight function sup_v∈ E ∫_0^1 e^(loge/s+γlogloge/s)v^2(s)/sds<∞, γ∈[0,1], where E:= {u ∈ H^1(0, 1): u(0) = 0, ∫_0^1 |u'|^2dx=1}. In particular, we prove compactness for all normalized sequences in the subcritical range γ<1, thereby resolving an open problem proposed in [J. M. do Ó, B. Ruf and P. Ubilla, CVPDE 2023]. We also study the critical Bliss-Moser inequality: nonumber sup_v∈ E I^c(v)< +∞, I^c(v)=∫_0^1 e^(loge/s+logloge/s+cs)v^2(s)/sds. For any c∈(−∞,+∞), we determine the sharp concentration level sup_v_j∈ E,v_j is NCSlimsup_j→∞ I^c(v_j)=1+e^2, and obtain explicit sufficient conditions for attainability under c>c^* and non-attainability under c<−c_*. This results describe how linear perturbations govern the transition between compactness, concentration, and attainability at the critical level.
ER -