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Revista Matemática Complutense

, Volume 29, Issue 2, pp 271–294 | Cite as

New results concerning Moser-type inequalities in Lorentz-Sobolev spaces

  • Robert ČernýEmail author
Article
  • 142 Downloads

Abstract

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(q\in (1,\infty ]\) and let \(\Omega \subset \mathbb {R}^n\) be an open bounded set. We obtain sharp constants concerning the Moser-type inequalities corresponding to the Lorentz-Sobolev space \(W_0^1L^{n,q}(\Omega )\) equipped with the norm
$$\begin{aligned} \Vert \nabla u\Vert _{(n,q)}:= {\left\{ \begin{array}{ll}\Vert t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{**}(t)\Vert _{L^q((0,\infty ))}&{}\text {for }q\in (1,\infty )\\ \sup _{t\in (0,\infty )}t^{\frac{1}{n}}|\nabla u|^{**}(t)&{}\text {for }q=\infty . \end{array}\right. } \end{aligned}$$
We also derive the key estimate for the Concentration-Compactness Principle in the case \(q\in (1,\infty )\) with respect to the above norm.

Keywords

Sobolev spaces Lorentz-Sobolev spaces Moser-Trudinger inequality Concentration-Compactness Principle Sharp constants 

Mathematics Subject Classification

46E35 46E30 26D10 

Notes

Acknowledgments

The author was supported by the ERC CZ Grant LL1203 of the Czech Ministry of Education. The author would like to thank Luboš Pick for fruitful discussions.

References

  1. 1.
    Adachi, S., Tanaka, K.: Trudinger type inequalities in \({\mathbb{R}}^N\) and their best exponents. Proc. Am. Math. Soc. 128(7), 2051–2057 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128, 385–398 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adimurthi, : Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the \(n\)-Laplacian. Ann. Sc. Norm. Sup. Pisa 17, 393–413 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Adimurthi, : Positive solutions of the semilinear Dirichlet problem with Critical growth in the unit disc in \({\mathbb{R}}^2\). Proc. Indian Acad. Sci. 99, 49–73 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Alvino, A., Ferone, V., Trombetti, Q.: Moser-type inequalities in Lorentz spaces. Potential Anal. 5, 273–299 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Černý, R.: Concentration-compactness principle for Moser-type inequalities in Lorentz-Sobolev spaces. Potential Anal. 43(1), 97–126 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Černý, R., Cianchi, A., Hencl, S.: Concentration-compactness principle for Moser-Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. 192(2), 225–243 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Černý, R., Gurka, P.: Moser-type inequalities for generalized Lorentz-Sobolev spaces. Houston. Math. J. 40(4), 1225–1269 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Černý, R., Mašková, S.: A sharp form of an embedding into multiple exponential spaces. Czechoslovak Math. J. 60(3), 751–782 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cianchi, A.: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems Indiana Univ. Math. J. 54, 669–705 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hencl, S.: A sharp form of an embedding into exponential and double exponential spaces. J. Funct. Anal. 204(1), 196–227 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hunt, R.A.: On \(L(p, q)\) spaces. Enseignement Math. 12(2), 249–276 (1966)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana 1(1), 145–201 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lorentz, G.G.: On the theory of spaces \(\Lambda \). Pac. J. Math. 1, 411–429 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Talenti, G.: Inequalities in rearrangement invariant function spaces. Nonlinear Anal. Funct. Spaces Appl. 5, 177–230 (1994)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Talenti, G.: An Inequality between \(u^*\), General Inequalities, 6 (Oberwolfach, 1990), Internat. Ser. Numer. Math, vol. 103, pp. 175–182. Birkhäuser, Basel (1992)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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