On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality \[ | | △ ϕ | | 2 ≥ S 2 | | ϕ | | 2 N N − 4 , N > 4 , ϕ ∈ D 0 2 , 2 ( R N ) , || φ ||_2 ≥ S_2 ||φ ||2N/N-4, N>4, φ ∈ {{ D}2,2₀({{ R}}^N)}, \] where S 2 S_2 is the best Sobolev constant and D 0 2 , 2 ( R N ) {{ D}2,2₀({{ R}}^N)} is the space obtained by taking the completion of C 0 <mml:mi mathv