Bounds for best constants in subcritical Sobolev embeddings

We establish upper and lower estimates for the embedding constants related to the classical Sobolev embeddings H01(Ω)↪Lp(Ω), where Ω⊆RN is a bounded domain or the whole RN and p∈(2,2N∕(N−2)), N≥3. In dimension N=2 we also derive the sharp asymptotic behavior of the embedding constant as p→∞.