Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs

Abstract: We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s\<1$, $p=1,2$, $1\leq q<p_{s}^{\ast}=\frac{Np}{N-sp}$ and $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain or the whole space $\mathbb{R}^{N}$. Our results cover the borderline case $p=1$, the Hilbert case $p=2$, $N\>2s$ and the so-called Sobolev limiting case $N=1$, $s=\frac{1}{2}$ and $p=2$, where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.