Bounds for subcritical best Sobolev constants in <i>W</i>^{1, <i>p</i>}

This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings\begin{document}$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q&lt; \begin{cases} \frac{Np}{N-p},&amp; 1\leq p&lt;N\\ \infty,&amp; p = N \end{cases} $\end{document}where \begin{document}$ N\geq p\geq1 $\end{document} and \begin{document}$ \Omega $\end{document} is a bounded smooth domain in \begin{document}$ \mathbb{R}^{N} $\end{document} or the whole space. The Sobolev limiting case \begin{document}$ p = N $\end{document} is also covered by means of a limiting procedure.