Advances in Nonlinear Analysis (Oct 2023)
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 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.
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