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

Abstract

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : inf { | u | 2 s ∗ | x | s ∫ Ω | ∇ u | 2 d x − λ 1 ∫ Ω u 2 | x − P 1 | 2 d x − λ 2 ∫ Ω u 2 | x − P 2 | 2 d x | u ∈ H 0 1 ( Ω ) , ∫ Ω | u | 2 s ∗ | x | s d x = 1 } where N ≥ 3 , Ω is a smooth domain, λ 1 , λ 2 ∈ R , 0 , P 1 , P 2 ∈ Ω , s ∈ ( 0 , 2 ) and 2 s ∗ = 2 ( N − s ) N − 2 . 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.