we complete the proof of Theorem \[Th3\] by proving the decay estimates . The proofs of these results rely again on our iteration scheme, this time applied with $\mathcal{I}_1=\emptyset$ and $\mathcal{I}_2$ being the set of all indices $i$ such that $p_i\le\overline{p}_0$ for some real number $\overline{p}_0$ (see ).

Finally, in Appendix \[App\], we prove a weak version of Kato’s inequality which is used in Sections \[Sec3\] and \[Sec4\].

[**Acknowledgments.**]{} The author wishes to express his gratitude to Frédéric Robert for helpful comments on the manuscript.

Application to the extremal functions of a class of\
anisotropic Sobolev inequalities {#Sec2}
====================================================

As mentioned in the introduction, one of our main motivation in this paper is to apply our results to the extremal functions of a class of anisotropic Sobolev inequalities which originates from Troisi [@Tro]. In this section, we first present in Theorem \[Th4\] below different equivalent versions of these inequalities, and we then prove in Theorem \[Th5\] that all these inequalities have extremal functions, and that with a suitable change of scale, these extremal functions are solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$.

We state the equivalent versions of the anisotropic Sobolev inequalities as follows.

\[Th4\] The following inequalities hold true.

1.  There exists 