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

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings W0s,p(Ω)↪Lq(Ω),{W}_{0}^{s,p}(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{q}(\Omega ), where N≥1N\ge 1, 02sN\gt 2s, and the so-called Sobolev limiting case N=1N=1, s=12s=\frac{1}{2}, and p=2p=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