On the stability of the Hardy–Sobolev inequality involving p-Laplace

This paper is devoted to considering the stability of the Hardy–Sobolev inequality ∫ R N | ∇ u | p d x ⩾ S β ( ∫ R N | u | p β ∗ | x | β d x ) p p β ∗ , for all u ∈ C c ∞ ( R N ) , with sharp constant S β > 0 , where 1 < p < N , 0 ⩽ β < p , p β ∗ = p ( N − β ) N − p . Firstly, since this problem involves the quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term | x | − β , some new estimates are established. Based on those works, we give the classification to the linearised problem related to the extremals then we investigate the gradient stability of above inequality by using spectral estimate combined with a compactness argument. Furthermore, we derive a remainder term inequality in the weak Lebesgue-norm sense for Ω ⊂ R N containing origin in it interior with | Ω | < ∞ , which reads that for all u ∈ C c ∞ ( Ω ) ∖ { 0 } , ‖ ∇ u ‖ L p ( Ω ) ( ∫ Ω | u | p β ∗ | x | β d x ) 1 p β ∗ − S β 1 p ⩾ C | Ω | − max { p , 2 } p 0 ∗ ( p − 1 ) ( ‖ u ‖ L w p 0 ∗ ( p − 1 ) / p ( Ω ) ‖ u ‖ L p 0 ∗ ( Ω ) ) max { p , 2 } for some constant C > 0 independent of Ω, where ‖ ⋅ ‖ L w s ( Ω ) denotes the weak L s -norm.