can write $$\label{Lem3Eq5}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)}\le d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}+r''\le d_{\operatorname{\overrightarrow{p}}}{\left(}x,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}+\frac{r''}{r}d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)},$$ and hence by putting together and , we obtain $$\label{Lem3Eq6}
d_{\operatorname{\overrightarrow{p}}}{\left(}x,0{\right)}\left|u{\left(}x{\right)}\right|^\delta\le\frac{r}{r-r''}\cdot K_1=\frac{2r}{r-r'}\cdot K_1.$$ By definition of $d_{\operatorname{\overrightarrow{p}}}$, then follows from . This proves our claim.

We prove by contradiction. Suppose that for any $\alpha\in{\mathbb{N}}$, there exists a Caratheodory function $f_\alpha:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ such that holds true, a solution $u_\alpha$ of with $f=f_\alpha$ such that $r_\kappa{\left(}u_\alpha{\right)}\le r'$ and $\left\|u_\alpha\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\le\kappa'$, and a point $x_\alpha\in{\mathbb{R}}^n\backslash B{\left(}0,r{\right)}$ such that $$\label{Lem3Eq7}
d_{\operatorname{\overrightarrow{p}}}{\left(}x_\alpha,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u_\alpha{\left(}x_\alpha{\right)}\right|^\delta>2\alpha\,.$$ It follows from and Poláčik–Quittner–Souplet [@PolQuiSou]\*[Lemma 5.1]{} that there exists $y_\alpha\in{\mathbb{R}}^n\backslash B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}$ such that $$\label{Lem3Eq8}
d_{\operatorname{\overrightarrow{p}}}{\left(}y_\alpha,B_{\operatorname{\overrightarrow{p}}}{\left(}0,r''{\right)}{\right)}\left|u_\alpha{\left(}y_\alpha{\right)}\right|^\delta>2\alpha\,,\quad\left|u_\alpha{\left(}x_\alpha{\right)}\right|\le\left|u_\alpha{\left(}y_\alpha{\right)}\right|,$$ and $$\label{Lem3Eq9}
\left|u_\alpha{\left(}y{\right)}\right|\le2^{1/\delta}\left|u_\alpha{\left(}y_\alpha{\right)}\right|\quad\text{for all }y\in B_{\operatorname{\overrightarrow{p}}}\big(y_\alpha,\alpha\,\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}\big).$$ For any $\alpha$ and $y\in{\mathbb{R}}^n$, we define $$\label{Lem3Eq10}
\widetilde{u}_\alpha{\left(}y{\right)}:=\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-1}\cdot u_\alpha{\left(}\tau_\alpha{\left(}y{\right)}{\right)},$$ where $$\tau_\alpha{\left(}y{\right)}:=y_\alpha+\big(\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{\frac{p_1-p^*}{p_1}}y_1,\dots,\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{\frac{p_n-p^*}{p_n}}y_n\big).$$ It follows from and that $$\label{Lem3Eq11}
\left|\widetilde{u}_\alpha{\left(}0{\right)}\right|=1\quad\text{and}\quad\left|\widetilde{u}_\alpha{\left(}y{\right)}\right|\le2^{1/\delta}\quad\text{for all }y\in B_{\operatorname{\overrightarrow{p}}}{\left(}0,\alpha{\right)}.$$ Moreover, by , we obtain $$\label{Lem3Eq12}
-\Delta_{\operatorname{\overrightarrow{p}}}\widetilde{u}_\alpha=\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{1-p^*}\cdot f_\alpha{\left(}\tau_\alpha{\left(}y{\right)},\left|u_\alpha{\left(}y_\alpha{\right)}\right|\cdot\widetilde{u}_\alpha{\right)}\quad\text{in }{\mathbb{R}}^n,$$ and gives $$\label{Lem3Eq13}
\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{1-p^*}\cdot\left|f_\alpha{\left(}\tau_\alpha{\left(}y{\right)},\left|u_\alpha{\left(}y_\alpha{\right)}\right|\cdot\widetilde{u}_\alpha{\right)}\right|\le\Lambda\left|\widetilde{u}_\alpha\right|^{p^*-1}.$$ By Lieberman’s gradient estimates [@Lie], it follows from and that there exists a constant $C>0$ such that for any $R>0$, we have $$\label{Lem3Eq14}
\left\|\nabla\widetilde{u}_\alpha\right\|_{L^\infty(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}\le C$$ for large $\alpha$. By Arzela–Ascoli Theorem and a diagonal argument, it follows from and that ${\left(}\widetilde{u}_\alpha{\right)}_\alpha$ converges up to a subsequence in $C^0_{\operatorname{loc}}{\left(}{\mathbb{R}}^n{\right)}$ to some Lipschitz continuous function $\widetilde{u}_\infty$ such that $\left|\widetilde{u}_\infty{\left(}0{\right)}\right|=1$. Moreover, by testing – with $\widetilde{u}_\alpha$, we obtain $$\label{Lem3Eq15}
\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\widetilde{u}_\alpha\right|^{p_i}dx\le\Lambda\int_{{\mathbb{R}}^n}\left|\widetilde{u}_\alpha\right|^{p^*}dx=\Lambda\int_{{\mathbb{R}}^n}\left|u_\alpha\right|^{p^*}dx\le\Lambda{\left(}\kappa'{\right)}^{p^*}.$$ Since $\left|\partial_{x_i}\left|\widetilde{u}_\alpha\right|\right|=\left|\partial_{x_i}\widetilde{u}_\alpha\right|$ a.e. in ${\mathbb{R}}^n$, 