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