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 W-0(s,p)(Omega) hooked right arrow L-q(Omega),where N >= 1 , 0 < s < 1 , p = 1,2 , 1 <= q < p(s )(& lowast;)= Np/N - sp , and Omega subset of R(N )is a bounded smooth domain or the whole space RN . 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 = (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.