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Maximizers for the singular Trudinger-Moser inequalities in the subcritical cases


Author: Nguyen Lam
Journal: Proc. Amer. Math. Soc. 145 (2017), 4885-4892
MSC (2010): Primary 35A23; Secondary 26D15, 46E35
DOI: https://doi.org/10.1090/proc/13624
Published electronically: June 9, 2017
MathSciNet review: 3692003
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Abstract: The main purpose of this note is to study the existence of extremal functions for the singular Trudinger-Moser inequalities in the subcritical cases. More precisely, let $N\geq 2$,$~0<\beta <N,~0<a,~b$ and denote \begin{align*} TM_{a,b,\beta }\left ( \alpha \right ) & =\sup _{\left \Vert \nabla u\right \Vert _{N}^{a}+\left \Vert u\right \Vert _{N}^{b}\leq 1}\int _{ \mathbb {R}^{N}}\phi _{N}\left ( \alpha \left ( 1-\frac {\beta }{N}\right ) \left \vert u\right \vert ^{\frac {N}{N-1}}\right ) \frac {dx}{\left \vert x\right \vert ^{\beta }},\\ \phi _{N}(t) & =e^{t}- {\displaystyle \sum \limits _{j=0}^{N-2}} \frac {t^{j}}{j!}. \end{align*} Then we will prove in this article that $TM_{a,b,\beta }\left ( \alpha \right )$ can be attained if ($\alpha <\alpha _{N}=N\omega _{N-1}^{\frac {1}{N-1}}$) or ($\alpha =\alpha _{N};$ $b<N$).


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Additional Information

Nguyen Lam
Affiliation: Department of Mathematics, University of British Columbia and The Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T1Z4, Canada
MR Author ID: 796424
ORCID: 0000-0002-8392-6284
Email: nlam@math.ubc.ca

Received by editor(s): August 25, 2016
Received by editor(s) in revised form: December 17, 2016
Published electronically: June 9, 2017
Additional Notes: The research of this work was partially supported by the PIMS-Math Distinguished Post-doctoral Fellowship from the Pacific Institute for the Mathematical Sciences.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2017 American Mathematical Society