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

Let Ω be a bounded domain of Rn (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 1−n/p) and the supercritical case (α<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|nA1(|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.