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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
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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

$\displaystyle TM_{a,b,\beta }\left ( \alpha \right )$ $\displaystyle =\sup _{\left \Vert \nabla u\right \Vert _{N}^{a}+\left \Vert u\r... ...vert ^{\frac {N}{N-1}}\right ) \frac {dx}{\left \vert x\right \vert ^{\beta }},$    
$\displaystyle \phi _{N}(t)$ $\displaystyle =e^{t}- {\displaystyle \sum \limits _{j=0}^{N-2}} \frac {t^{j}}{j!}.$    

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
Email: nlam@math.ubc.ca

DOI: https://doi.org/10.1090/proc/13624
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