$\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded, where $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}$ is as in , we have $$\begin{gathered}
\label{Lem4Eq1}
\left\|u\right\|_{L^\gamma{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}{\right)}}^\gamma\le c_0\gamma^{p^*}\max_{i\in\mathcal{I}_1\cup\mathcal{I}_2}\Big(\min{\left(}1,\gamma-p_*+1{\right)}^{-p_i}{\left(}\lambda'-\lambda{\right)}^{-p_i}q_i^{p_i}\\
\times R_{\delta_i}^{-\frac{p_i}{q_i}}\left\|u\right\|_{L^{\gamma_i}{\left(}\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}{\right)}}^{\gamma_i}\Big)^\frac{n}{n-p},\end{gathered}$$ where $\delta_i:=1$ if $i\in\mathcal{I}_1$, $\delta_i:=2$ if $i\in\mathcal{I}_2$, and $\gamma_i:=\frac{n-p}{n}\gamma+p_i-p$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$.

Preliminary to the proof of Lemma \[Lem4\], we prove the following result.

\[Lem5\] Let $v$ be a nonnegative solution in $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$ of $$\label{Lem5Eq1}
-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}\quad\text{in }{\mathbb{R}}^n,$$ for some real number $\Lambda>0$, where the inequality must be understood in the weak sense as in . Let $\beta>-1$ and $\eta\in C^1{\left(}{\mathbb{R}}^n{\right)}$ be such that $0\le\eta\le1$ in ${\mathbb{R}}^n$, $\eta v$ has compact support, and $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, where $p_-:=\min{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$ and $p_+:=\max{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. Then there exists a constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$ such that $$\begin{gathered}
\label{Lem5Eq2}
{\left(}\int_{{\mathbb{R}}^n}{\left(}\eta v{\right)}^{\frac{n{\left(}\beta+p{\right)}}{n-p}}dx{\right)}^{\frac{n-p}{n}}\le C{\left(}\left|\beta\right|^p+1{\right)}\bigg(\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_-}v^{\beta+p^*}dx\\
+\sum_{i=1}^n\min{\left(}1,\beta+1{\right)}^{-p_i}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\eta\right|^{p_i}\eta^{\beta+p_--p_i}v^{\beta+p_i}dx\bigg).\end{gathered}$$

The finiteness of the integrals in is ensured by the fact that $v\in L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\eta v$ has compact support, and $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$.

For any $\varepsilon>0$, we define $v_\varepsilon:=v+\varepsilon\overline\eta$, where $\overline\eta$ is a cutoff function on a neighborhood of the support of $\eta v$ such that $\overline\eta^{\beta+1}\in C^1{\left(}{\mathbb{R}}^n{\right)}$. Since $v\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}\cap L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, and ${\left(}\beta+p_-{\right)}/p_+\le{\left(}\beta+p_-{\right)}/p_-=1+\beta/p_-$, we get ${\left(}\eta v_\varepsilon{\right)}^{\min{\left(}1,1+\beta/p_-{\right)}}\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. By a generalized version of the anisotropic Sobolev inequality (see Cîrstea–Vétois [@CirVet]\*[Lemma A.1]{}), we then obtain $$\label{Lem5Eq3}
\left\|\eta v_\varepsilon\right\|_{L^{\frac{n{\left(}\beta+p{\right)}}{n-p}}{\left(}{\mathbb{R}}^n{\right)}}^{\beta+p}\le C{\left(}\beta+p{\right)}^p\prod_{i=1}^n\left\|{\left(}\eta v_\varepsilon{\right)}^{\frac{\beta}{p_i}}\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right\|_{L^{p_i}{\left(}{\mathbb{R}}^n{\right)}}^{\frac{p}{n}}<\infty$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. For any $i=1,\dotsc,n$, we 