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 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