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On the Hardy–Sobolev–Maz'ya Inequality and Its Generalizations

  • Yehuda Pinchover
  • Kyril Tintarev
Part of the International Mathematical Series book series (IMAT, volume 8)

Abstract

Abstract The paper deals with natural generalizations of the Hardy— Sobolev—Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.

Keywords

Sobolev Inequality Potential Term Hardy Inequality Null Sequence Positive Continuous Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Allegretto, M., Huang, Y.X.: A Picone's identity for the p-Laplacian and applications. Nonlinear Anal., Theory Methods Appl. 32, 819–830 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Allegretto, M., Huang, Y.X.: Principal eigenvalues and Sturm comparison via Picone's identity. J. Differ. Equations 156, 427–438 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benguria, R., Frank, R., Loss, M.: The Sharp Constant in the Hardy–Sobolev—Maz'ya Inequality in the Three Dimensional Upper Half-Space. Preprint, arXiv:0705.3833Google Scholar
  4. 4.
    Brezis, H., Marcus, M.: Hardy's inequalities revisited. Dedicated to Ennio De Giorgi. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 25, 217–237 (1997)MATHMathSciNetGoogle Scholar
  5. 5.
    Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weight. Compos. Math. 53, 259–275 (1984)MATHMathSciNetGoogle Scholar
  6. 6.
    del Pino, M., Elgueta, M., Manasevich, R.: A homotopic deformation along p of a Leray—Schauder degree result and existence for (ǁu′ǁp−2u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p> 1. J. Differ. Equations 80, 1–13 (1989)MATHCrossRefGoogle Scholar
  7. 7.
    Diaz, J.I., Saá, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci., Paris Ser. I Math. 305, 521–524 (1987)MATHMathSciNetGoogle Scholar
  8. 8.
    Filippas, S., Maz'ya, V., Tertikas, A.: Critical Hardy–Sobolev inequalities. J. Math. Pures Appl. (9) 87, 37–56 (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Filippas, S., Tertikas, A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Filippas, S., Tertikas, A., Tidblom, J.: On the Structure of Hardy–Sobolev—Maz'ya Inequalities. Preprint arXiv:0802.0986Google Scholar
  11. 11.
    Fleckinger-Pellé, J., Hernández, J., Takáč, P., de Thélin, F.: Uniqueness and positivity for solutions of equations with the p-Laplacian. In: Proc. Conf. Reaction-Diffusion Equations (Trieste, 1995), pp.141–155. Lect. Notes Pure Appl. Math. 194, Marcel Dekker, New York (1998)Google Scholar
  12. 12.
    García-Melián, J., Sabina de Lis, J.: Maximum and comparison principles for operators involving the p-Laplacian. J. Math. Anal. Appl. 218, 49–65 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Il'in, V.P.: Some integral inequalities and their applications in the theory of differen-tiable functions of several variables (Russian). Mat. Sb. 54, no. 3, 331–380 (1961)MathSciNetGoogle Scholar
  14. 14.
    Mancini, G., Sandeep, K.: On a Semilinear Elliptic Equation in Hn. Preprint (2007)Google Scholar
  15. 15.
    Marcus, M., Shafrir, I.: An eigenvalue problem related to Hardy's L p inequality. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 29, 581–604 (2000)MATHMathSciNetGoogle Scholar
  16. 16.
    Maz'ya, V.: Sobolev Spaces. Springer-Verlag, Berlin (1985)Google Scholar
  17. 17.
    Pinchover, Y., Tertikas, A., Tintarev, K.: A Liouville-type theorem for the p-Laplacian with potential term. Ann. Inst. H. Poincaré. Anal. Non Linéaire 25, 357–368 (2008)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pinchover, Y., Tintarev, K.: Ground state alternative for singular Schrödinger operators. J. Funct. Anal. 230, 65–77 (2006)MATHMathSciNetGoogle Scholar
  19. 19.
    Pinchover, Y., Tintarev, K.: Ground state alternative for p-Laplacian with potential term. Calc. Var. Partial Differ. Equ. 28 (2007), 179–201MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Pinchover, Y., Tintarev, K.: On positive solutions of minimal growth for singular p-Laplacian with potential term. Adv. Nonlin. Studies 8, 213–234 (2008)MATHMathSciNetGoogle Scholar
  21. 21.
    Smets, D.: Nonlinear Schrödinger equations with Hardy potential and critical non-linearities. Trans. Am. Math. Soc. 357, 2909–2938 (2005)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Takáč, P., Tintarev, K.: Generalized minimizer solutions for equations with the p-Laplacian and a potential term. Proc. R. Soc. Edinb., Sect. A, Math. 138, 201–221 (2008)MATHGoogle Scholar
  23. 23.
    Tertikas, A., Tintarev, K.: On existence of minimizers for the Hardy–Sobolev—Maz'ya inequality. Ann. Mat. Pura Appl. (4) 183, 165–172 (2004)MathSciNetGoogle Scholar
  24. 24.
    Troyanov, M.: Parabolicity of manifolds. Sib. Adv. Math. 9, 125–150 (1999)MATHMathSciNetGoogle Scholar
  25. 25.
    Troyanov, M.: Solving the p-Laplacian on manifolds. Proc. Am. Math. Soc. 128, 541–545 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Yehuda Pinchover
    • 1
  • Kyril Tintarev
    • 2
  1. 1.Technion — Israel Institute of TechnologyHaifaIsrael
  2. 2.Uppsala UniversityUppsalaSweden

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