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Potential Analysis

, Volume 43, Issue 1, pp 97–126 | Cite as

Concentration-Compactness Principle for Moser-type Inequalities in Lorentz-Sobolev Spaces

  • Robert Černý
Article

Abstract

Let n𝜖, n ≥ 2, q ∈ (1, ) and let \({\Omega }\subset \mathbb {R}^{n}\) be an open bounded set. We study the Concentration-Compactness Principle for the embedding of the Lorentz-Sobolev space \({W_{0}^{1}}L^{n,q}({\Omega })\) into an Orlicz space corresponding to a Young function \(t\mapsto \exp (t^{q^{\prime }})-1\). The results are stated with respect to the (quasi-)norm
$$||\nabla u||_{n,q}:= ||t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{*}(t)||_{L^{q}((0,|{\Omega}|))} \ . $$

Keywords

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

Mathematics Subject Classification (2010)

46E35 46E30 26D10 

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References

  1. 1.
    Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbb {R}^{N}\) and their best exponents. Proc. Amer. Math. Soc. 128(7), 2051–2057 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. 128, 385–398 (1988)MATHMathSciNetCrossRefGoogle 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)Google 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)Google Scholar
  5. 5.
    Alberico, A.: Moser Type Inequalities for Higher-Order Derivatives in Lorentz Spaces. Potential Anal. 28, 389–400 (2008)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Alvino, A., Ferone, V., Trombetti, Q.: Moser-Type Inequalities in Lorentz Spaces. Potential Anal. 5, 273–299 (1996)MATHMathSciNetGoogle Scholar
  7. 7.
    Cerny, 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)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cerny, R., Maskova, S.: A sharp form of an embedding into multiple exponential spaces. Czechoslovak Math. J. 60(3), 751–782 (2010)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chabrowski, J.: Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var. 3(4), 493–512 (1995)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cianchi, A.: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54, 669–705 (2005)MATHMathSciNetCrossRefGoogle 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)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hunt, R.A.: On L(p, q) spaces. Enseignement Math. 12(2), 249–276 (1966)MATHMathSciNetGoogle 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)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lorentz, G.G.: On the theory of spaces Λ. Pacific J. Math. 1, 411–429 (1951)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRefGoogle Scholar
  16. 16.
    Talenti, G.: An inequality between u and ∇u General Inequalities, 6 (Oberwolfach, 1990), Internat. Ser. Numer. Math., Birkhuser, Basel, vol. 103, pp. 175–182 (1992)Google Scholar
  17. 17.
    Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

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

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