Missing terms in the weighted Hardy-Sobolev inequalities and its application

摘要

Let Ω be a bounded domain of R~n (n ≥ 1) containing the origin. In the present paper we establish the weighted Hardy-Sobolev inequalities with sharp remainders. For example, when α = 1 - n/p and 1 < p < +oo hold, we establish the following inequality. There exist positive numbers Λn,p,α,C, and R such that we have ∫_Ω|▽u|~p|x|~(αp)dx≥∧n,p,a∫_Ω|u(x)|~p/|x|~nA_1(|x|)~-pdx(0.1)+C∫_Ω|u(x)|~p/|x|~nA_1(|x|)~-pA_2(|x|)~-2dx for any u ∈ ~(1,p)_(α,0)(Ω). Here A_1(|x|) = logR/|x|, and A_2(|x|) = logA_1(|x|). This is called the critical Hardy-Sobolev inequality with a sharp remainder involving a singular weight A_1(|x|)~-pA_2(|x|)~-2, in the sense that the improved inequality holds for this weight but fails for any weight more singular than this one. Here Λn,p,α is a sharp constant independent of each function u. Further we establish the Hardy-Sobolev inequalities in the subcritical case (a > 1 - n/p) and the supercritical case (a < 1 - n/p). As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by -div(|x|~αp|▽u|~p-2|▽u)-μ/|x|~nA_1(|x|)~-p|u|~p-2u (the critical case) will tend to zero as μ increases to Λn,p,α. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.