Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : (Formula presented.) where (Formula presented.), Ω is a smooth domain, (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems.