Absolute continuity of the best Sobolev constant

Let λ q ≔ inf { ‖ ∇ u ‖ L p ( Ω ) p / ‖ u ‖ L q ( Ω ) p : u ∈ W 0 1 , p ( Ω ) ∖ { 0 } } be the best Sobolev constant of the immersion W 0 1 , p ( Ω ) ↪ L q ( Ω ) , where Ω is a bounded and smooth domain of R N , 1 < p < N and 1 ≤ q ≤ p ⋆ ≔ N p N − p . We prove that the function q ∈ [ 1 , p ⋆ ] ⟼ λ q is absolutely continuous.