## Abstract

Let Ω be a smooth bounded domain in ℝ^{N} with 0 ∈ Ω and let p ∈ (1,∞)\{N}. By a classical inequality of Hardy we have ∫Ω|∇v|^{p}>c_{p,N}^{*}∫ Ω|v|^{p}/|x|^{p}, for all 0≠v ∈ W_{0}^{1,p}(Ω\{0}), with c_{p,N}^{*}=|(N-p)/p|^{p} being the best constant in this inequality. More generally, for η ∈ C(Ω̄) such that η≥0,η≠0 and η(0)=0 we have, for certain values of λ, that ∫Ω|∇v|^{p}-λη|v|^{p}/|x|^{p} >c_{p,N}^{*}∫Ω|v|^{p}/|x|^{p} , for all 0≠v ∈ W_{0}^{1,p}(Ω\{0}). In particular, it follows that there is no minimizer for this inequality. We consider then a family of approximating problems, namelyinf0≠v ∈ W_{0}^{1,p}(Ω\{0})∫Ω|∇v|^{p}- λη|v|^{p}/|x|^{p}∫Ω|v|^{p-ε}/| x|^{p}for ε>0, and study the asymptotic behavior, as ε→0, of the positive minimizers {u_{ε}} which are normalized by ∫Ωu_{ε}^{p}=1. We prove the convergence u_{ε}→u_{*} in ∩_{1<q<p}W_{0}^{1,q}(Ω\{0}), where u_{*} is the unique positive solution (up to a multiplicative factor) of the equation -Δ_{p}u=(u^{p-1}/|x|^{p})(c_{p,N}^{ *}+λη(x)) in Ω\{0}, with u=0 on ∂Ω.

Original language | English |
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Pages (from-to) | 1221-1240 |

Number of pages | 20 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 54 |

Issue number | 7 |

DOIs | |

State | Published - 1 Sep 2003 |

Externally published | Yes |

## Keywords

- Hardy's inequality
- P-Laplacian
- Singular elliptic problem

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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