Absolute continuity of the best Sobolev constant of a bounded domain

Let $\lambda_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(\Omega)}^{p}/\Vertu\Vert_{L^{q}(\Omega)}^{p}:u\in W_{0}^{1,p}(\Omega)\setminus{0}} $,where$\Omega$ is a bounded and smooth domain of ${\mathbb{R}}^N,$ $1<p<N$ and $1\leq q\leq p^{\star}% :=\frac{Np}{N-p}.$Weprovethatthefunction$q\mapsto\lambda_{q}$ is absolutely continuous in the closed interval $[1,p^{\star }]\mathrm{.}$