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Remarks on the behavior of the best Sobolev constants

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Contributions to Nonlinear Elliptic Equations and Systems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 86))

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Abstract

We study the function

$$\displaystyle{q \in [1,p^{\star })\mapsto \lambda _{ q}(\varOmega ) =\inf \left \{\left \Vert \nabla u\right \Vert _{L^{p}(\varOmega )}^{p}: u \in W_{ 0}^{1,p}(\varOmega ),\mbox{ }\left \Vert u\right \Vert _{ L^{q}(\varOmega )} = 1\right \},}$$

where p > 1, Ω is a bounded domain of \(\mathbb{R}^{N},\) N ≥ 2 and \(p^{\star }\) is the critical exponent of the Sobolev immersion \(W_{0}^{1,p}(\varOmega )\hookrightarrow L^{q}(\varOmega )\).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30.123-970, Brazil

    Grey Ercole

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  1. Grey Ercole
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Correspondence to Grey Ercole .

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Editors and Affiliations

  1. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Alexandre Nolasco de Carvalho

  2. Dipartimento di Matematica, Università degli Studi di Milano, Lombardia, Milano, Italy

    Bernhard Ruf

  3. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Ederson Moreira dos Santos

  4. Département de Mathématique. CP 214, Université Libre de Bruxelles, Brussels, Belgium

    Jean-Pierre Gossez

  5. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Sergio Henrique Monari Soares

  6. Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, Paris, France

    Thierry Cazenave

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Ercole, G. (2015). Remarks on the behavior of the best Sobolev constants. In: Nolasco de Carvalho, A., Ruf, B., Moreira dos Santos, E., Gossez, JP., Monari Soares, S., Cazenave, T. (eds) Contributions to Nonlinear Elliptic Equations and Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 86. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19902-3_12

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