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Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality

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Abstract

We prove a concentration-compactness principle for the Trudinger-Moser functional associated with a class of weighted Sobolev spaces including fractional dimensions. Based in this result and using blow up analysis we establish a sharp form of Trudinger-Moser type inequality for this class of weighted Sobolev spaces. Moreover, we discuss the existence of extremal for the maximizing problem associated with this new Trudinger-Moser inequality.

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Acknowledgments

Research 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 141853/2012-3.

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Author notes

  1. José Francisco de Oliveira

    Present address: Department of Mathematics, Federal University of Piauí, 64049-550 , Teresina, PI, Brazil

Authors and Affiliations

  1. Department of Mathematics, Federal University of Pernambuco, 50740-540 , Recife, PE, Brazil

    José Francisco de Oliveira

  2. Department of Mathematics, Federal University of Paraíba, 58051-900 , João Pessoa, PB, Brazil

    João Marcos do Ó

Authors
  1. José Francisco de Oliveira
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  2. João Marcos do Ó
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Correspondence to João Marcos do Ó.

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Communicated by A. Chang.

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de Oliveira, J.F., do Ó, J.M. Concentration-compactness principle and extremal functions for a sharp Trudinger-Moser inequality. Calc. Var. 52, 125–163 (2015). https://doi.org/10.1007/s00526-014-0707-z

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