Absolute continuity of the best Sobolev constant

Let λq≔inf{‖∇u‖Lp(Ω)p/‖u‖Lq(Ω)p:u∈W01,p(Ω)∖{0}} be the best Sobolev constant of the immersion W01,p(Ω)↪Lq(Ω), where ΩΩ is a bounded and smooth domain of RN,1