Archiv der Mathematik

, Volume 100, Issue 3, pp 273–287 | Cite as

A weighted Hardy inequality and nonexistence of positive solutions



In this article, we prove that the following weighted Hardy inequality
$$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$
holds with optimal Hardy constant \({\big(\frac{|d-p|}{p}\big)^{p}}\) for all \({u \in W^{1,p}_{\mu,0}(\Omega)}\) if the dimension d ≥ 2, 1 < p < d, and for all \({u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}\) if p > d ≥ 1. Here we assume that Ω is an open subset of \({\mathbb{R}^{d}}\) with \({0 \in \Omega}\) , A is a real d × d-symmetric positive definite matrix, c > 0, and
$$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$
If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on \({W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}\) . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.

Mathematics Subject Classification (2010)

Primary 35A01 35B09 35B25 35D30 35K67 35K92 


Weighted Hardy’s inequality p-Kolmogorov operator Singular perturbation Nonexistence 


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  1. 1.
    Aguilar Crespo J.A., Peral Alonso I.: Global behavior of the Cauchy problem for some critical nonlinear parabolic equations. SIAM J. Math. Anal. 31, 1270–1294 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121–139.Google Scholar
  3. 3.
    H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983, Théorie et applications. [Theory and applications].Google Scholar
  4. 4.
    X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 973–978.Google Scholar
  5. 5.
    DiBenedetto E.: Degenerate parabolic equations, Universitext. Springer- Verlag, New York (1993)CrossRefGoogle Scholar
  6. 6.
    V. A. Galaktionov, On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations, Tr. Mat. Inst. Steklova 260 (2008), 123–143.Google Scholar
  7. 7.
    J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476.Google Scholar
  8. 8.
    G. R. Goldstein, J. A. Goldstein, and A. Rhandi, Kolmogorov equations perturbed by an inverse-square potential, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 623–630.Google Scholar
  9. 9.
    G. R. Goldstein, J. A. Goldstein, and A. Rhandi, Weighted Hardy’s inequality and the Kolmogorov equation perturbed by an inverse-square potential, Applicable Analysis, doi: 10.1080/00036811.2011.587809 (2011), 1–15.
  10. 10.
    Goldstein J.A., Kombe I.: Nonlinear degenerate prabolic equations with singular lower-order term. Adv. Differential Equations 8, 1153–1192 (2003)MathSciNetMATHGoogle Scholar
  11. 11.
    Hardy G.H.: Note on a theorem of Hilbert. Math. Z. 6, 314–317 (1920)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki 67 (2000), no. 4, 563–572.Google Scholar
  13. 13.
    D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991.Google Scholar
  14. 14.
    M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984, Corrected reprint of the 1967 original.Google Scholar
  15. 15.
    Sturm C.: Mémoir sur une classe d’équations à différences partielles. J. Math. Pures Appl. 1, 373–444 (1836)Google Scholar
  16. 16.
    J. M. Tölle, Uniqueness of weighted Sobolev spaces with weakly differentiable weights, to appear in J. Funct. Anal. (in press) (2012), 23.Google Scholar
  17. 17.
    Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear diffusion equations, World Scientific Publishing Co. Inc., River Edge, NJ, 2001, Translated from the 1996 Chinese original and revised by the authors.Google Scholar

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Authors and Affiliations

  1. 1.Institut für Angewandte Analysis, Universität UlmUlmGermany
  2. 2.Université de Lorraine (campus de Metz) et CNRSLaboratoire de Mathématiques et Applications de MetzMetz Cedex 1France
  3. 3.Dipartimento di MatematicaUniversità degli Studi di SalernoFisciano (Sa)Italy

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