Absolute continuity of the best Sobolev constant of a bounded domain

Let $λ_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(Ω)}^{p}/\Vertu\Vert_{L^{q}(Ω)}^{p}:u\in W_{0}^{1,p}(Ω)\setminus{0}} $, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N},$ $1<p<N$ and $1\leq q\leq p^{\star}$ We prove that the function $q\mapstoλ_{q}$ is absolutely continuous in the closed interval $[1,p^{\star}].$

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