A sharp Trudinger-Moser type inequality in $\mathbb {R}^2$
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- by Manassés de Souza and João Marcos do Ó PDF
- Trans. Amer. Math. Soc. 366 (2014), 4513-4549 Request permission
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.References
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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
- MR Author ID: 365349
- Email: jmbo@pq.cnpq.br
- 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 - © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 4513-4549
- MSC (2010): Primary 35J20; Secondary 35J60
- DOI: https://doi.org/10.1090/S0002-9947-2014-05811-X
- MathSciNet review: 3217691