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Peer-reviewedCC BY 4.0Mathematics

Improved Moser–Trudinger Inequalities with Singular Weights and Nonlinear Perturbations for Zero-Mean Functions

  • Geijuan Gong1
  • Yueli Liu1
  • Xin Li1
  1. 1School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China
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Abstract

We study Moser–Trudinger inequalities with singular weights and nonlinear perturbations for zero-mean functions on a smooth bounded domain Ω⊂R2 with 0∈∂Ω. We consider supu∈H1(Ω), ∥∇u∥22=1∫Ωudx=0∫Ωe2π(1−β)u2q(∥u∥22)|x|2βdx, where 0<β<1 and q(t)=1+a1t+⋯+aktk has nonnegative coefficients. Under the cascading conditions 0≤a1<λ and 0≤aj≤λaj−1 (j=2,…,k), where λ is the first nonzero Neumann eigenvalue, we prove that the supremum is finite. Our results unify and extend the improved Moser–Trudinger inequality of Lu–Yang for zero-mean functions and the singular Moser–Trudinger inequality of Zhu.

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Cite

BibTeXRIS
BibTeX
@article{gong2026improved,
  title = {{Improved Moser–Trudinger Inequalities with Singular Weights and Nonlinear Perturbations for Zero-Mean Functions}},
  author = {Geijuan Gong and Yueli Liu and Xin Li},
  journal = {Mathematics},
  year = {2026},
  month = nov,
  doi = {10.3390/math14214106},
  url = {https://www.mdpi.com/2227-7390/14/21/4106},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Improved Moser–Trudinger Inequalities with Singular Weights and Nonlinear Perturbations for Zero-Mean Functions
AU  - Geijuan Gong
AU  - Yueli Liu
AU  - Xin Li
PY  - 2026
DA  - 2026/11/09
JO  - Mathematics
DO  - 10.3390/math14214106
UR  - https://www.mdpi.com/2227-7390/14/21/4106
AB  - We study Moser–Trudinger inequalities with singular weights and nonlinear perturbations for zero-mean functions on a smooth bounded domain Ω⊂R2 with 0∈∂Ω. We consider supu∈H1(Ω), ∥∇u∥22=1∫Ωudx=0∫Ωe2π(1−β)u2q(∥u∥22)|x|2βdx, where 0<β<1 and q(t)=1+a1t+⋯+aktk has nonnegative coefficients. Under the cascading conditions 0≤a1<λ and 0≤aj≤λaj−1 (j=2,…,k), where λ is the first nonzero Neumann eigenvalue, we prove that the supremum is finite. Our results unify and extend the improved Moser–Trudinger inequality of Lu–Yang for zero-mean functions and the singular Moser–Trudinger inequality of Zhu.
SN  - 2227-7390
ER  -