Bounds for subcritical best Sobolev constants in W1, p

Abstract

<p style='text-indent:20px;'>This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ N\geq p\geq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> or the whole space. The Sobolev limiting case <inline-formula><tex-math id="M5">\begin{document}$ p = N $\end{document}</tex-math></inline-formula> is also covered by means of a limiting procedure.</p>