construct quasi-explicit examples of solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ for anisotropic configurations of type $\operatorname{\overrightarrow{p}}={\left(}p_-,\dotsc,p_-,p_+,\dotsc,p_+{\right)}$ by using the method of separation of variables (see Vétois [@Vet1]). These solutions turn out to vanish in the $i$-th directions corresponding to $p_i=p_+$, exactly like what we prove to be true in Theorem \[Th3\] in case $p_*<p_+<p^*$.

The paper is organized as follows. In Section \[Sec2\], we present different equivalent versions of the anisotropic Sobolev inequality, and we study the existence and scaling properties of extremal functions for these inequalities.

Section \[Sec3\] is concerned with preliminary properties satisfied by the solutions of , namely global boundedness results and a weak decay estimate.

In Sections \[Sec4\] and \[Sec5\], we perform a Moser-type iteration scheme inspired from the one developed in Cîrstea–Vétois [@CirVet] for the fundamental solutions. In order to treat a large part of the proofs in a unified way, we consider a general family of domains defined as $$\label{Eq6}
\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}:=\bigg\{x\in{\mathbb{R}}^n:\sum_{i\in\mathcal{I}_1}\left|x_i\right|^{q_i}\hspace{-2pt}<\hspace{-2pt}{\left(}1+\lambda{\right)}R_1\text{ and }\bigg|\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{q_i}-R_2\bigg|\hspace{-2pt}<\hspace{-2pt}\lambda R_2\bigg\},$$ where $\lambda\in{\left(}0,1{\right)}$, $R_1,R_2>0$, $\mathcal{I}_1$ and $\mathcal{I}_2$ are two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ is such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. On these domains, we prove that the solutions of satisfy reverse Hölder-type inequalities of the form $$\label{Eq7}
\left\|u\right\|_{L^\gamma{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}{\right)}}^\gamma\le C\max_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\Big({\left(}\lambda'-\lambda{\right)}^{-p_i}\\
R_{\delta_i}^{-\frac{p_i}{q_i}}\left\|u\right\|_{L^{\gamma_i}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}{\right)}}^{\gamma_i}\Big)^\frac{n}{n-p}$$ 