In particular, for any minimizer $u$ of , since $\mu_{\operatorname{\overrightarrow{p}},u}=1$ and $\rho_{\operatorname{\overrightarrow{p}},u}=\tau_{\operatorname{\overrightarrow{p}}}$, it follows from – that $u\circ\tau_{\operatorname{\overrightarrow{p}}}^{-1}$ is an extremal function of .

Next, we prove that the extremal functions of do not change sign. We let $C_0$ be the best constant and $u$ be an extremal function of . By writing $u=u_+-u_-$, where $u_+:=\max{\left(}u,0{\right)}$ and $u_-:=\max{\left(}-u,0{\right)}$, we obtain $$\begin{gathered}
\label{Th5Eq5}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx={\left(}\frac{1}{C_0}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p_*}={\left(}\frac{1}{C_0}\int_{{\mathbb{R}}^n}u_-^{p^*}dx+\frac{1}{C_0}\int_{{\mathbb{R}}^n}u_+^{p^*}dx{\right)}^{p/p_*}\\
\le{\left(}{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_-\right|^{p_i}dx{\right)}^{p^*/p}+{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u_+\right|^{p_i}dx{\right)}^{p^*/p}{\right)}^{p/p_*}.\end{gathered}$$ It follows from that either $u_-=0$ or $u_+=0$, and hence we obtain that the function $u$ has constant sign.

Finally, from the Euler-Lagrange equation satisfied by $u$, namely $$-\sum_{i=1}^np_i\partial_{x_i}{\left(}\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u{\right)}=\lambda{\left(}u{\right)}\left|u\right|^{p^*-2}u,\quad\text{where}\quad\lambda{\left(}u{\right)}:=\frac{\overset{n}{\underset{i=1}{\sum}}p_i\int_{{\mathbb{R}}^n}\big|\partial_{x_i}u\big|^{p_i}dx}{\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx}\,,$$ we derive that the function $x\in{\mathbb{R}}^n\mapsto\mu u{\left(}\mu_1x_1,\dotsc,\mu_nx_n{\right)}$ with $\mu_i:={\left(}\lambda{\left(}u{\right)}/p_i{\right)}^{1/p_i}$ for all $i=1,\dotsc,n$ is a solution of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$. This ends the proof of Theorem \[Th5\].

Preliminary results {#Sec3}
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From now on, we are concerned with the general case of an arbitrary solution of .

For any $s\in{\left(}0,\infty{\right)}$ and any domain $\Omega\subset{\mathbb{R}}^n$, we define the weak Lebesgue space $L^{s,\infty}{\left(}\Omega{\right)}$ as the set of all measurable functions $u:\Omega\to{\mathbb{R}}$ such that $$\left\|u\right\|_{L^{s,\infty}{\left(}\Omega{\right)}}:=\sup_{h>0}\big(h\cdot\operatorname{meas}{\left(}\left\{\left|u\right|>h\right\}{\right)}^{1/s}\big)<\infty\,,$$ where $\operatorname{meas}{\left(}\left\{\left|u\right|>h\right\}{\right)}$ is the measure of the set $\left\{x\in\Omega:\,\left|u{\left(}x{\right)}\right|>h\right\}$. The map $\left\|\cdot\right\|_{L^{s,\infty}{\left(}\Omega{\right)}}$ defines a quasi-norm on $L^{s,\infty}{\left(}\Omega{\right)}$. We refer, for instance, to the book of Grafakos [@Gra] for the material on weak Lebesgue spaces.

The first result 