Noncritical Weighted Hardy's Inequalities with compact perturbations

Let $\mathrm{\Omega }$ be a bounded domain of ${\mathbb{R}}^N$ whose boundary is a ${\mathbb{C}}^2$ compact manifolds. In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we adopted powers of the distance function $\delta (x)$ to the boundary $\mathrm{\partial }\mathrm{\Omega }$.