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A sharp Trudinger-Moser type inequality in $ \mathbb{R}^2$


Authors: Manassés de Souza and João Marcos do Ó
Journal: Trans. Amer. Math. Soc. 366 (2014), 4513-4549
MSC (2010): Primary 35J20; Secondary 35J60
Published electronically: May 7, 2014
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Abstract: In this paper, we establish a sharp Trudinger-Moser type inequality for a class of Schrödinger operators in $ \mathbb{R}^2$. We obtain a result related to the compactness of the embedding of a subspace of $ W^{1,2}(\mathbb{R}^2)$ into the Orlicz space $ L_{\phi }(\mathbb{R}^2)$ determined by $ \phi (t)=e^{\beta t^{2}}-1$. Our result is similar to one obtained by Adimurthi and Druet for smooth bounded domains in $ \mathbb{R}^2$, which is closely related to a compactness result proved by Lions. Furthermore, similarly to what has been done by Carleson and Chang, we prove the existence of an extremal function for this Trudinger-Moser inequality by performing a blow-up analysis. Trudinger-Moser type inequalities have a wide variety of applications to the study of nonlinear elliptic partial differential equations involving the limiting case of Sobolev inequalities and have received considerable attention in recent years.


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Additional Information

Manassés de Souza
Affiliation: Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife, PE, Brazil
Email: mxs@dmat.ufpe.br

João Marcos do Ó
Affiliation: Departamento de Matemática, Universidade Federal de Paraíba, 58051-900 João Pessoa, PB, Brazil
Email: jmbo@pq.cnpq.br

DOI: https://doi.org/10.1090/S0002-9947-2014-05811-X
Keywords: Trudinger-Moser inequality, blow-up analysis, extremal function
Received by editor(s): June 22, 2011
Received by editor(s) in revised form: February 16, 2012
Published electronically: May 7, 2014
Additional Notes: The authors’ research was partially supported by the National Institute of Science and Technology of Mathematics INCT-Mat, CAPES and CNPq grants 307400/2009-3, 620108/2008-8 and 142002/2006-2.
The second author was the corresponding author
Article copyright: © Copyright 2014 American Mathematical Society