Let H 1(R 2)denote the Sobolev space of all real valued functions on R 2 which, together with their gradients, are square integrable. Let $$\begin{gathered} \lambda \equiv \inf \iint\limits_{R^2 } {|\nabla \varphi |^2 dxdy(\iint\limits_{R^2 } {\varphi ^2 dxdy/}\iint\limits_{R^2 } {\varphi ^6 dxdy})^{1/2} \equiv \inf J_2 (\varphi )} \hfill \\ \varphi \in H^1 (R^2 ),\varphi \ne 0 \hfill \\ \end{gathered} $$ 1 weakly convergent subsequence and that if at least one weak limit is nonzero then there is a function ϕ 0 eH 1(R 2)such that J 2(ϕ 0)=λand that λ∼π4/3.We provide as asymptotic formula for ϕ 0 namely, we show ϕ 0(r)≈K0(r)· ·[K 0 4 (r)+1]−1/2 as r→+∞,where K 0 is the modified Bessel function of the second kind. An extension to the more general inequality $$C = \inf (\int\limits_{R^m } {|\nabla \varphi |^p dx} )^{\alpha /p} (\int\limits_{R^m } {|\varphi |^q dx} )^{\beta /q} (\int\limits_{R^m } {|\varphi |^r dx} )^{ - 1/r} $$ , β>0,α+β= 1and r −1=βq−1+α(p −1−m −1),p, q, r>1is briefly discussed.