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 $1\leq q\leq p^{\star}$ We prove that the function $q\mapsto\lambda_{q}$ is absolutely continuous in the closed interval $[1,p^{\star}].$