On minimization problems which approximate Hardy Lp inequality Academic Article uri icon

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( Ω ) 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, namely inf 0≠v∈W 0 1,p (Ω⧹{0}) ∫ Ω | ∇ v| p −λ η|v| p /|x| p ∫ Ω |v| p−e /|x| p for e >0, and study the asymptotic behavior, as e →0, of the positive minimizers { u e } which are normalized by ∫ Ω u e p =1 . We prove the convergence u e →u ∗ in ⋂ 1 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 .

publication date

  • January 1, 2003