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Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension $ N$


Author: Van Hoang Nguyen
Journal: Proc. Amer. Math. Soc. 147 (2019), 5183-5193
MSC (2010): Primary 46E35, 26D10
DOI: https://doi.org/10.1090/proc/14566
Published electronically: August 28, 2019
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Abstract: We provide a simpler proof of the Moser-Trudinger type inequality with logarithmic weight $ w_\beta = (-\ln \vert x\vert)^{\beta (N-1)}$, $ \beta \in [0,1)$ in dimension $ N\geq 2$ recently established by Calanchi and Ruf. Our proof is based on a suitable change of functions on $ B$ and the classical Moser-Trudinger inequality on $ B$. We also prove the existence of maximizers for this inequality when $ \beta \geq 0$ sufficiently small.


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Additional Information

Van Hoang Nguyen
Affiliation: Institute of Research and Development, Duy Tan University, Da Nang, Vietnam
Email: vanhoang0610@yahoo.com; and nguyenvanhoang14@duytan.edu.vn

DOI: https://doi.org/10.1090/proc/14566
Keywords: Moser--Trudinger type inequality, concentration level, extremal function, logarithmic weight
Received by editor(s): March 27, 2017
Received by editor(s) in revised form: November 20, 2018, and January 27, 2019
Published electronically: August 28, 2019
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society