a constant $C=C\big(n,\operatorname{\overrightarrow{p}}\big)$ such that $$\label{Th4Eq1}
    {\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{n/p^*}\le C\prod_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{1/p_i}\quad\forall u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}.$$

2.  For any $\overrightarrow{\theta}={\left(}\theta_1,\dotsc,\theta_n{\right)}$ such that $\theta_i>0$ for all $i=1,\dotsc,n$ and $\sum_{i=1}^n1/\theta_i=n/p$, there exists a constant $C_{\overrightarrow{\theta}}=C\big(n,\operatorname{\overrightarrow{p}},\overrightarrow{\theta}\big)$ such that $$\label{Th4Eq2}
    {\left(}\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx{\right)}^{p/p^*}\le C_{\overrightarrow{\theta}}\sum_{i=1}^n{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{\theta_i/p_i}\quad\forall u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}.$$ In particular, we get in case $\theta_i=p_i$ for all $i=1,\dotsc,n$.

As a remark, the inequalities and enjoy an anisotropic scaling law. Indeed, it can easily be seen that every integral in these inequalities are invariant with respect to the change of scale $u\mapsto u_\lambda$, where $$\label{Eq9}
u_\lambda{\left(}x{\right)}=\lambda u{\left(}\lambda^{{\left(}p^*-p_1{\right)}/p_1}x_1,\dotsc,\lambda^ {{\left(}p^*-p_n{\right)}/p_n}x_n{\right)}$$ for all $\lambda>0$ and $x\in{\mathbb{R}}^n$.

We refer to Troisi [@Tro]\*[Theorem 1.2]{} for the proof of the inequality . Then the inequality follows from by applying an inequality of weighted arithmetic and geometric means. As a remark, we can also obtain from by applying the change of scale below.

Regarding the extremal functions of and , we prove the following result. The existence part in this result will be obtained as a corollary of the work of El Hamidi–Rakotoson [@ElHRak] and Proposition \[Pr1\] below.

\[Th5\] If $p_+<p^*$, then there exist extremal functions $u\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$, $u\ne0$, of and . Moreover, for any extremal function $u$ of or , there exist $\mu_1,\dotsc,\mu_n>0$ such that the function $x\in{\mathbb{R}}^n\mapsto u{\left(}\mu_1x_1,\dotsc,\mu_nx_n{\right)}$ is a 