all $a,b>0$. Moreover, Cordero-Erausquin–Nazaret–Villani [@CorNazVil] proved that the functions are the only extremal functions of . In case where the norm of the gradient in is replaced bv the Euclidean norm, the existence of radially symmetric extremal functions was found by Aubin [@Aub], Rodemich [@Rod], and Talenti [@Tal]. Since $p/{\left(}p_*-p{\right)}={\left(}n-p{\right)}/{\left(}p-1{\right)}$, the decay rate in coincides with the one in .

In case of the Laplace operator ($p_i=2$), Caffarelli–Gidas–Spruck [@CafGidSpr] (see also Chen–Li [@ChenLi]) proved that every positive solution of with $f{\left(}x,u{\right)}=u^{p^*-1}$ is of the form . This result can be extended to the case where all $p_i$ are equal to $p\in{\left(}1,n{\right)}$ for positive solutions satisfying the one-dimensional symmetry $u{\left(}x{\right)}=u\big(\sum_{i=1}^n\left|x_i\right|^{\frac{p}{p-1}}\big)$ for all $x\in{\mathbb{R}}^n$. Indeed, this result has been proved by Guedda–Véron [@GueVer] in case of positive, radially symmetric solutions for the $p$–Laplace equation $-\operatorname{div}{\left(}\left|\nabla u\right|^{p-2}\nabla u{\right)}=u^{p^*-1}\quad\text{in }{\mathbb{R}}^n$, and it can easily be seen that both cases lead to the same ordinary differential equation. We also mention that radial symmetry results have been established for positive solutions in $D^{1,p}{\left(}{\mathbb{R}}^n{\right)}$ in the case of $p$–Laplace equations (see Damascelli–Merchán–Montoro–Sciunzi [@DamMerMonSci], Damascelli–Ramaswamy [@DamRam], Sciunzi [@Sci], and Vétois [@Vet3]).

Theorem \[Th1\] has been proved in Vétois [@Vet3] in case of the $p$–Laplace operator. We also refer 