have $$\label{Lem5Eq4}
\int_{{\mathbb{R}}^n}{\left(}\eta v_\varepsilon{\right)}^\beta\left|\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right|^{p_i}dx\le2^{p_i-1}{\left(}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\eta\right|^{p_i}\eta^\beta v_\varepsilon^{\beta+p_i}dx+\int_{{\mathbb{R}}^n}\left|\partial_{x_i} v_\varepsilon\right|^{p_i}\eta^{\beta+p_i}v_\varepsilon^\beta dx{\right)}.$$ Since $v\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}\cap L^\infty{\left(}{\mathbb{R}}^n{\right)}$, $\overline\eta^{\beta+1}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, $\eta^{{\left(}\beta+p_-{\right)}/p_+}\in C^1{\left(}{\mathbb{R}}^n{\right)}$, and ${\left(}\beta+p_-{\right)}/p_+\le\beta+p_-$, we get $\eta^{\beta+p_i}v_\varepsilon^{\beta+1}\in D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. For any $i=1,\dotsc,n$, since $v_\varepsilon\equiv v+\varepsilon$ on the support of $\eta v$, testing with $\eta^{\beta+p_i}v_\varepsilon^{\beta+1}$ gives $$\begin{gathered}
\label{Lem5Eq5}
{\left(}\beta+1{\right)}\sum_{j=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^\beta dx\le\Lambda\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
-{\left(}\beta+p_i{\right)}\sum_{j=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j-2}{\left(}\partial_{x_j}v{\right)}{\left(}\partial_{x_j}\eta{\right)}\eta^{\beta+p_i-1}v_\varepsilon^{\beta+1}dx\,.\end{gathered}$$ For any $i,j=1,\dotsc,n$, Youngs inequality yields $$\begin{gathered}
\label{Lem5Eq6}
-{\left(}\beta+p_i{\right)}\left|\partial_{x_j}v\right|^{p_j-2}{\left(}\partial_{x_j}v{\right)}{\left(}\partial_{x_j}\eta{\right)}\eta^{\beta+p_i-1}v_\varepsilon^{\beta+1}\\
\le\frac{p_j-1}{p_j}\cdot{\left(}\beta+1{\right)}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^{\beta}+\frac{1}{p_j}\cdot\frac{{\left(}\beta+p_i{\right)}^{p_j}}{{\left(}\beta+1{\right)}^{p_j-1}}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}.\end{gathered}$$ It follows from and that $$\begin{gathered}
\label{Lem5Eq7}
\sum_{j=1}^n\frac{1}{p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}v\right|^{p_j}\eta^{\beta+p_i}v_\varepsilon^\beta dx\le\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
+\sum_{j=1}^n\frac{1}{p_j}\cdot{\left(}\frac{\beta+p_i}{\beta+1}{\right)}^{p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}dx\,.\end{gathered}$$ In particular, by and , we obtain $$\begin{gathered}
\label{Lem5Eq8}
\int_{{\mathbb{R}}^n}{\left(}\eta v_\varepsilon{\right)}^\beta\left|\partial_{x_i}{\left(}\eta v_\varepsilon{\right)}\right|^{p_i}dx\le C\bigg(
\Lambda{\left(}\beta+1{\right)}^{-1}\int_{{\mathbb{R}}^n}\eta^{\beta+p_i}v^{p^*-1}v_\varepsilon^{\beta+1}dx\\
+\sum_{j=1}^n\min{\left(}1,\beta+1{\right)}^{-p_j}\int_{{\mathbb{R}}^n}\left|\partial_{x_j}\eta\right|^{p_j}\eta^{\beta+p_i-p_j}v_\varepsilon^{\beta+p_j}dx+\varepsilon^{\beta+p_i}\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\overline\eta\right|^{p_i}\eta^{\beta+p_i}\overline\eta^{\beta}dx\bigg)\end{gathered}$$ for some constant $C=C{\left(}n,\operatorname{\overrightarrow{p}}{\right)}$. Finally, since $\eta^{p_i}\le\eta^{p_-}$, we get by plugging into and passing to the limit as $\varepsilon\to0$. This ends the proof of Lemma \[Lem5\].

Now, we can prove Lemma \[Lem4\] by using Lemma \[Lem5\].

We denote $\beta:=\frac{n-p}{n}\gamma-p$. In particular, $\gamma>p_*-1$ is equivalent to $\beta>-1$. In connexion with the sets $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)}$, we define test functions of the form $$\label{Lem4Eq2}
\eta{\left(}x{\right)}:=\bigg[\overline\eta_{\lambda,\lambda'}\bigg(R_1^{-1}\sum_{i\in\mathcal{I}_1}\left|x_i\right|^{q_i}\bigg)\widetilde\eta_{\lambda,\lambda'}\bigg(R_2^{-1}\sum_{i\in\mathcal{I}_2}\left|x_i\right|^{q_i}\bigg)\bigg]^{\max{\left(}1,\frac{p_+}{\beta+p_-}{\right)}}$$ for all $x\in{\mathbb{R}}^n$, where $\overline\eta_{\lambda,\lambda'},\,\widetilde\eta_{\lambda,\lambda'}\in C^1{\left(}0,\infty{\right)}$ satisfy $0\le\overline\eta_{\lambda,\lambda'},\,\widetilde\eta_{\lambda,\lambda'}\le1$ in ${\left(}0,\infty{\right)}$, $\overline\eta_{\lambda,\lambda'}=1$ in ${\left[}0,1+\lambda{\right]}$, $\overline\eta_{\lambda,\lambda'}=0$ in ${\left[}1+\lambda',\infty{\right)}$, $\left|\overline\eta'_{\lambda,\lambda'}\right|\le2$ in ${\left[}1+\lambda,1+\lambda'{\right]}$, $\widetilde\eta_{\lambda,\lambda'}=1$ in ${\left[}1-\lambda,1+\lambda{\right]}$, $\widetilde\eta_{\lambda,\lambda'}=0$ in ${\left[}0,1-\lambda'{\right]}\cup{\left[}1+\lambda',\infty{\right)}$, and $\left|\widetilde\eta'_{\lambda,\lambda'}\right|\le2/{\left(}\lambda'-\lambda{\right)}$ in ${\left[}1-\lambda',1-\lambda{\right]}\cup{\left[}1+\lambda,1+\lambda'{\right]}$. With these properties of $\overline\eta_{\lambda,\lambda'}$ and $\widetilde\eta_{\lambda,\lambda'}$, we obtain $$0\le\eta\le1\text{ in }{\mathbb{R}}^n,\,\eta=1\text{ in }\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda{\right)},\text{ and }\eta=0\text{ in }{\mathbb{R}}^n\backslash \Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}.$$ Since $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap\operatorname{supp}{\left(}u{\right)}$ is bounded by assumption, we get that $\eta u$ has compact 