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Archiv der Mathematik

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

A weighted Hardy inequality and nonexistence of positive solutions

  • Daniel Hauer
  • Abdelaziz Rhandi
Article

Abstract

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 

Keywords

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

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Copyright information

© Springer Basel 2013

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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