Advertisement

Sharp Adams-type inequality invoking Hardy inequalities

  • Mohamed Khalil Zghal
Article
  • 49 Downloads

Abstract

We establish a sharp Trudinger–Moser type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of [8].

Keywords

Trudinger–Moser inequalities Hardy inequalities Orlicz space lack of compactness 

Mathematics Subject Classification

46E35 35B33 46E30 

Notes

Acknowledgements

The author is very grateful to Prof. Hajer Bahouri and Prof. Mohamed Majdoub for interesting discussions and careful reading of the manuscript. A part of this work was done at LAMA-Université Paris-Est Créteil whose hospitality and support is gratefully acknowledged.

References

  1. 1.
    Adachi S and Tanaka K, Trudinger type inequalities in \(\mathbb{R}^N\) and their best exponents, Proc. Am. Math. Soc. 128 (1999) 2051–2057CrossRefzbMATHGoogle Scholar
  2. 2.
    Adams D R, A sharp inequality of J. Moser for higher order derivates, Ann. Math. 128 (1988) 385–398MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bahouri H, Structure theorems for 2D linear and nonliner Schrödinger equations, Commun. Contemp. Math. 10 (2015) 1142Google Scholar
  4. 4.
    Bahouri H, Structure theorems for 2D linear and nonliner Schrödinger equations, Notes C.R. Acad. Sci. Paris 353 (2015) 235–240zbMATHGoogle Scholar
  5. 5.
    Bahouri H, Chemin J-Y and Gallagher I, Refined Hardy inequalities, Ann. Scuola Norm. Pisa, 5 (2006) 375–391MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bahouri H and Cohen A, Refined Sobolev inequalities in Lorentz spaces, J. Fourier Anal. Appl. 17 (2011) 662–673MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bahouri H, Ibrahim S and Perelman G, Scattering for the critical 2-D NLS with exponential growth, J. Differ. Integral Equ. 27 (2014) 233–268MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bahouri H, Majdoub M and Masmoudi N, On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal. 260 (2011) 208–252MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bahouri H, Majdoub M and Masmoudi N, Lack of compactness in the 2D critical Sobolev embedding, the general case, J. Math. Pures Appl. 101 (2014) 415–457MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bahouri H and Perelman G, A Fourier approach to the profile decomposition in Orlicz spaces, Math. Res. Lett. 21 (2014) 33–54MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ben Ayed I and Zghal M K, Characterization of the lack of compactness of \(H^2_{{\rm rad}}({\mathbb{R}}^4)\) into the Orlicz space, Commun. Contemp. Math. 16 (2014)Google Scholar
  12. 12.
    Ben Ayed I and Zghal M K, Description of the lack of compactness in Orlicz spaces and applications, J. Differ. Integral Equ. 28 (2015) 553–580MathSciNetzbMATHGoogle Scholar
  13. 13.
    Colliander J, Ibrahim S, Majdoub M and Masmoudi N, Energy critical NLS in two space dimensions, J. Hyperbol. Differ. Equ. 6(3) (2009) 549–575MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hardy G H, Note on a theorem of Hilbert, Math. Z. 6 (1920) 314–317MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hardy G H, An inequality between integrals, Messenger of Math. 54 (1925) 150–156Google Scholar
  16. 16.
    Ibrahim S, Majdoub M and Masmoudi N, Global solutions for a semilinear, two-dimensional Klein–Gordon equation with exponential-type nonlinearity, Commun. Pure Appl. Math. (2006) 1–20Google Scholar
  17. 17.
    Ibrahim S, Majdoub M, Masmoudi N and Nakanishi K, Scattering for the two-dimentional energy-critical wave equation, Duke Math. J. 150 (2009) 287–329MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lam N, and Lu G, Sharp Adams inequalities in Sobolev spaces \(W^{m,\frac{n}{m}}({\mathbb{R}}^n)\) for arbitrary integer \(m\), J. Differ. Equ. 253(4) (2012) 1143–1171CrossRefzbMATHGoogle Scholar
  19. 19.
    Lions PL, The concentration-compactness principle in the calculus of variations. The limit case. I., Rev. Mat. Iberoam. 1 (1985) 145–201MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lions P-L, The concentration-compactness principle in the calculus of variations. The limit case. II., Rev. Mat. Iberoam. 1 (1985) 45–121MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Masmoudi N and Sani F, Adams’ Inequality with the exact growth condition in \({\mathbb{R}}^4\), Commun. Pure Pure Appl. Math. (2014) 1307–1335Google Scholar
  22. 22.
    Moser J, A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J. 20 (1971) 1077–1092MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pohozaev S I, The Sobolev embedding in the case \(pl=n\), Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Sections, Moscov. Energet. Inst., Moscow, (1965) 158–170Google Scholar
  24. 24.
    Rao M M, and Ren Z D, Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250 (2002)Google Scholar
  25. 25.
    Ruf B, A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb{R}^2\), J. Funct. Anal. 219 (2005) 340–367MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ruf B, and Sani F, Sharp Adams-type inequalities in \(\mathbb{R}^n\), Trans. Am. Math. Soc. 365 (2013) 645–670MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ruf B, and Sani F, Ground states for elliptic equations in \({\mathbb{R}}^2\) with exponential critical growth, Geom. Prop. Parabol. Elliptic PDE’s, 2 (2013) 251–267CrossRefzbMATHGoogle Scholar
  28. 28.
    Sani F, A biharmonic equation in \({\mathbb{R}}^4\) involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal. 12 (2013) 251–267MathSciNetGoogle Scholar
  29. 29.
    Sani F, A biharmonic equation in \({\mathbb{R}}^4\) involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud. 11 (2011) 889–904MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Trudinger N S, On imbeddings into Orlicz spaces and some applications, J. Appl. Math. Mech. 17 (1967) 473–484MathSciNetzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Laboratoire Équations aux dérivées partielles (LR03ES04), Faculté des Sciences de TunisUniversité de Tunis El ManarTunisTunisia

Personalised recommendations