}x\in{\mathbb{R}}^n,$$ for some real number $\Lambda>0$. Here, $p^*$ denotes the critical Sobolev exponent and is defined as $$p^*:=\frac{n}{\sum_{i=1}^n\frac{1}{p_i}-1}=\frac{np}{n-p}\qquad\text{with}\quad\frac{1}{p}:=\frac{1}{n}\sum_{i=1}^n\frac{1}{p_i}\,.$$

The problem with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ appears in the study of extremal functions for a class of anisotropic Sobolev inequalities. Early references on anisotropic Sobolev inequalities are Nikolski[ĭ]{} [@Nik], Troisi [@Tro], and Trudinger [@Tru]. We also refer to Cianchi [@Cia1] for a more recent work on the topic. Here we are interested in an inequality which appeared first in Troisi [@Tro]. Among different equivalent versions (see Theorem \[Th4\] below), this inequality can be stated as $$\label{Eq3}
\int_{{\mathbb{R}}^n}\left|u\right|^{p^*}dx\le C{\left(}\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}u\right|^{p_i}dx{\right)}^{p^*/p}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$ and for all functions $u\in C^\infty_c{\left(}{\mathbb{R}}^n{\right)}$. The inequality enjoys an anisotropic scaling law (see below). As a corollary of the work of El Hamidi–Rakotoson [@ElHRak], we obtain in Theorem \[Th5\] below that there exist extremal functions for the inequality provided that $p_i<p^*$ for all $i=1,\dotsc,n$.

In the presence of anisotropy, namely when the $p_i$ are not all equal, there is no explicit formula for the extremal functions of . This motivates to find a priori estimates for these functions, and more generally for the solutions of equations of type . The main difficulties in this work come from the non-homogeneity of the 