On best constants in Hardy inequalities with a remainder term

Let Ω be a bounded open set of RN containing the origin. We compute the best value of the constant C(α,|Ω|) in ∫Ω|∇u|2dx−(N−2)42∫Ωu2|x|2dx≥C(α,|Ω|)‖u‖L(2NN−α,2)2, with α<2 and u∈H01(Ω). Then we get the optimal value of C(|Ω|) in ∫Ω|∇u|3dx−(N−33)3∫Ωu3|x|3dx≥C(|Ω|)‖u‖L33, where u∈W01,3(Ω).