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 W 0 s , p ( Ω ) ↪ L q ( Ω ) , {W}_{0}^{s,p}(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{q}(\Omega ), where N ≥ 1 N\ge 1 , 0 < s < 1 0\lt s\lt 1 , p = 1 , 2 p=1,2 , 1 ≤ q < p s ∗ = N p N − s p 1\le q\lt {p}_{s}^{\ast }=\frac{Np}{N-sp} , and Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded smooth domain or the whole space R N {{\mathbb{R}}}^{N} . Our results cover the borderline case p = 1 p=1 , the Hilbert case p = 2 p=2 , N > 2 s N\gt 2s , and the so-called Sobolev limiting case N = 1 N=1 , s = 1 2 s=\frac{1}{2} , and p = 2 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.