Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials,we consider the following minimization problem : inf{integral(Omega) |del u|(2) dx-lambda 1 integral(Omega) u(2)|x-P1|(2) dx -lambda(2)integral(Omega) u(2)|x-P-2|(2) dx| u is an element of H-0(1)(Omega),integral(Omega) |u|(2)|x|(s) dx = 1} whereN >= 3,is a smooth domain,lambda(1),lambda(2) is an element of R,0,P-1,P-2 is an element of Omega,s is an element of(0, 2) and 2(s)(*) = 2(N-s)N-2. Concerning the coefficients of Hardy potentials, we derive asharp threshold for the existence and non-existence of a minimizer. In addi-tion, we study the existence and non-existence of a positive solution to theEuler-Lagrangian equations corresponding to the minimization problems