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Summary Let H1(R2)denote the Sobolev space of all real valued functions on R2 which, together with their gradients, are square integrable. Let$$\begin{gathered}  \lambda  \equiv\inf\iint\limits_{R^2 } {|\nabla \varphi |^2 dxdy(\iint\limits_{R^2 } {\varphi ^2 dxdy/}\iint\limits_{R^2 } {\varphi ^6 dxdy})^{1/2}  \equiv \inf J_2 (\varphi )} \hfill \\  \varphi  \in H^1 (R^2 ),\varphi  \ne 0 \hfill \\ \end{gathered} $$1weakly convergent subsequence and that if at least one weak limit is nonzero then there is a function ?0 eH1(R2)such that J2(?0)=?and that ?~p4/3.We provide as asymptotic formula for ?0namely, we show ?0(r)˜K0(r)· ·[K04(r)+1]-1/2as r?+8,where K0 is the modified Bessel function of the second kind. An extension to the more general inequality$$C = \inf  (\int\limits_{R^m } {|\nabla \varphi |^p dx} )^{\alpha /p} (\int\limits_{R^m } {|\varphi |^q dx} )^{\beta /q} (\int\limits_{R^m } {|\varphi |^r dx} )^{ - 1/r} $$, ß>0,a+ß= 1and r-1=ßq-1+a(p-1-m-1),p, q, r>1is briefly discussed.

An estimate for the best constant in a Sobolev inequality involving three integral norms