On minimization problems which approximate Hardy Lp inequality

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2 Scopus citations

Abstract

Let Ω be a smooth bounded domain in ℝN with 0 ∈ Ω and let p ∈ (1,∞)\{N}. By a classical inequality of Hardy we have ∫Ω|∇v|p>cp,N*∫ Ω|v|p/|x|p, for all 0≠v ∈ W01,p(Ω\{0}), with cp,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 >cp,N*∫Ω|v|p/|x|p , for all 0≠v ∈ W01,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 ∈ W01,p(Ω\{0})∫Ω|∇v|p- λη|v|p/|x|p∫Ω|v|p-ε/| x|pfor ε>0, and study the asymptotic behavior, as ε→0, of the positive minimizers {uε} which are normalized by ∫Ωuεp=1. We prove the convergence uε→u* in ∩1<q<pW01,q(Ω\{0}), where u* is the unique positive solution (up to a multiplicative factor) of the equation -Δpu=(up-1/|x|p)(cp,N *+λη(x)) in Ω\{0}, with u=0 on ∂Ω.

Original languageEnglish
Pages (from-to)1221-1240
Number of pages20
JournalNonlinear Analysis, Theory, Methods and Applications
Volume54
Issue number7
DOIs
StatePublished - 1 Sep 2003
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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