it follows from that ${\left(}\left|\widetilde{u}_\alpha\right|{\right)}_\alpha$ converges weakly up to a subsequence to $\left|\widetilde{u}_\infty\right|$ in $D^{1,\operatorname{\overrightarrow{p}}}{\left(}{\mathbb{R}}^n{\right)}$. Passing to the limit into –, we then obtain that $\left|\widetilde{u}_\infty\right|$ is a weak solution of the inequality $$\label{Lem3Eq16}
-\Delta_{\operatorname{\overrightarrow{p}}}\left|\widetilde{u}_\infty\right|\le\Lambda\left|\widetilde{u}_\infty\right|^{p^*-1}\quad\text{in }{\mathbb{R}}^n.$$ In particular, since $\left|\widetilde{u}_\infty{\left(}0{\right)}\right|=1$, it follows from Lemma \[Lem2\] that $\left\|u_\infty\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\ge\kappa_0$, and hence there exists a real number $R>0$ such that $$\label{Lem3Eq17}
\left\|u_\infty\right\|_{L^{p^*}{\left(}B{\left(}0,R{\right)}{\right)}}>\kappa\,.$$ On the other hand, we have $$\label{Lem3Eq18}
\left\|\widetilde{u}_\alpha\right\|_{L^{p^*}(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}=\left\|u_\alpha\right\|_{L^{p^*}{\left(}B_{\operatorname{\overrightarrow{p}}}{\left(}y_\alpha,R\cdot\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}{\right)}{\right)}}.$$ By and since $r_\kappa{\left(}u_\alpha{\right)}<r''$, we obtain $$\label{Lem3Eq19}
B_{\operatorname{\overrightarrow{p}}}\big(y_\alpha,R\cdot\left|u_\alpha{\left(}y_\alpha{\right)}\right|^{-\delta}\big)\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,r_\kappa{\left(}u_\alpha{\right)}{\right)}=\emptyset$$ for large $\alpha$. By definition of $r_\kappa{\left(}u_\alpha{\right)}$, it follows from and that $$\label{Lem3Eq20}
\left\|\widetilde{u}_\alpha\right\|_{L^{p^*}(B_{\operatorname{\overrightarrow{p}}}{\left(}0,R{\right)})}\le\kappa$$ for large $\alpha$, which is in contradiction with . This ends the proof of Lemma \[Lem3\].

The reverse Hölder-type inequalities {#Sec4}
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The following result is a key step in the Moser-type iteration scheme that we develop in the next section.

\[Lem4\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true, $u$ be a solution of , and $\kappa$, $r$, and $K_0$ be as in Lemma \[Lem3\]. Let $\mathcal{I}_1$ and $\mathcal{I}_2$ be two disjoint subsets of $\left\{1,\dotsc,n\right\}$, $\mathcal{I}_2\ne\emptyset$, and ${\overrightarrow{q}}={\left(}q_i{\right)}_{i\in\mathcal{I}_1\cup\mathcal{I}_2}$ be such that $q_i>1$ for all $i\in\mathcal{I}_1\cup\mathcal{I}_2$. Then there exists a constant $c_0=c_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,K_0{\right)}>1$ such that for any $R_1,R_2>0$, $\lambda<\lambda'\in{\left(}0,1/2{\right]}$, and $\gamma>p_*-1$ such that $\Omega_{{\overrightarrow{q}}}{\left(}\mathcal{I}_1,R_1,\mathcal{I}_2,R_2,\lambda'{\right)}\cap B_{\operatorname{\overrightarrow{p}}}{\left(}0,\max{\left(}r,1{\right)}{\right)}=\emptyset$ and 