function $v$, and applying the anisotropic Sobolev inequality, we obtain $$\label{Lem2Eq}
\Lambda\int_{{\mathbb{R}}^n}v^{p^*}dx\ge\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}v\right|^{p_i}dx\ge K{\left(}\int_{{\mathbb{R}}^n}v^{p^*}dx{\right)}^{\frac{n-p}{n}}$$ for some constant $K=K{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. The result then follows from with $\kappa_0:={\left(}K/\Lambda{\right)}^{\frac{n-p}{p^2}}$.

As a last result in this section, we prove the following decay estimate. This result is not sharp, but it turns out to be a crucial ingredient in what follows.

\[Lem3\] Assume that $p_+<p^*$. Let $\kappa_0$ be as in Lemma \[Lem2\], $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, and $u$ be a solution of . For any $\kappa>0$, we define $$\label{Lem3Eq1}
r_\kappa{\left(}u{\right)}:=\inf\big(\big\{r>0\,:\,\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}{\right)}}<\kappa\big\}\big),$$ where $B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}$ is the open ball of center 0 and radius $r$ with respect to the distance function $d_{\operatorname{\overrightarrow{p}}}$ defined as $$\label{Lem3Eq2}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,y{\right)}:=\sum_{i=1}^n\left|x_i-y_i\right|^{\frac{\delta p_i}{p^*-p_i}}\quad\text{with}\quad\delta:=\frac{p^*-p_+}{p_+},$$ for all $x,y\in{\mathbb{R}}^n$. Then for any $\kappa\in{\left(}0,\kappa_0{\right)}$ and $r>r_\kappa{\left(}u{\right)}$, there exists a constant $K_0=K_0\big(n,\operatorname{\overrightarrow{p}},\Lambda,\kappa,r,r_\kappa{\left(}u{\right)},\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\big)$ such that $$\label{Lem3Eq3}
\left|u{\left(}x{\right)}\right|\le K_0\bigg(\sum_{i=1}^n\left|x_i\right|^{\frac{p_i}{p^*-p_i}}\bigg)^{-1}\quad\text{for all }x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}.$$

This proof is adapted from Vétois [@Vet3]\*[Lemma 3.1]{} We fix $\Lambda>0$, $\kappa\in{\left(}0,\kappa_0{\right)}$, $\kappa'>\kappa_0$, $r>0$, and $r'\in{\left(}0,r{\right)}$. We claim that in order to obtain Lemma \[Lem3\], it is sufficient to prove that there exists a constant $K_1=K_1{\left(}n,\operatorname{\overrightarrow{p}},\kappa,\kappa',r,r'{\right)}$ such that for any solution $u$ of such that $r_\kappa{\left(}u{\right)}\le r'$ and $\left\|u\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\le\kappa'$, we have $$\label{Lem3Eq4}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u{\left(}x{\right)}\right|^\delta\le K_1\quad\text{for all }x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)},$$ where $r'':={\left(}r+r'{\right)}/2$. Indeed, for any $x\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r{\right)}$, we 