in this section is as follows.

\[Lem1\] Assume that $p_+<p^*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true. Then any solution of belongs to $W^{1,\infty}{\left(}{\mathbb{R}}^n{\right)}\cap L^{p_*-1,\infty}{\left(}{\mathbb{R}}^n{\right)}$, and hence by interpolation, to $L^s{\left(}{\mathbb{R}}^n{\right)}$ for all $s\in{\left(}p_*-1,\infty{\right]}$.

The $L^\infty$–boundedness of the solutions follows from a straightforward adaptation of El Hamidi–Rakotoson [@ElHRak]\*[Propositions 1 and 2]{}, the first proposition being in turn adapted from Fragalà–Gazzola–Kawohl [@FraGazKaw]\*[Theorem 2]{}.

Once we have the $L^\infty$–boundedness of the solutions, we obtain the $L^\infty$–boundedness of the derivatives by applying Lieberman’s gradient estimates [@Lie].

The proof of the $L^{p_*-1,\infty}$–boundedness of the solutions follows exactly the same arguments as in Vétois [@Vet3]\*[Lemma 2.2]{}. One only has to replace $\left|\nabla u\right|^p$ by $\sum_{i=1}^n\left|\partial_{x_i}u\right|^{p_i}$.

For any solution $u$ of , by Proposition \[Pr2\] in Appendix \[App\], we obtain $$\label{Eq10}
-\Delta_{\operatorname{\overrightarrow{p}}}\left|u\right|\le f{\left(}x,u{\right)}\cdot\operatorname{sgn}{\left(}u{\right)}\le\Lambda\left|u\right|^{p^*-1}\quad\text{in }{\mathbb{R}}^n,$$ where $\operatorname{sgn}{\left(}u{\right)}$ denotes the sign of $u$ and the inequality is in the sense that for any nonnegative, smooth function $\varphi$ with compact support in ${\mathbb{R}}^n$, we have $$\sum_{i=1}^n\int_{{\mathbb{R}}^n}\left|\partial_{x_i}\left|u\right|\right|^{p_i-2}{\left(}\partial_{x_i}\left|u\right|{\right)}{\left(}\partial_{x_i}\varphi{\right)}dx\le\Lambda\int_{{\mathbb{R}}^n}\left|u\right|^{p^*-1}\varphi\,dx\,.$$

We prove the following result.

\[Lem2\] For any real number $\Lambda>0$ and any nonnegative, nontrivial solution $v\in D^{1,p}{\left(}{\mathbb{R}}^n{\right)}$ of the inequality $-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}$ in ${\mathbb{R}}^n$, we have $\left\|v\right\|_{L^{p^*}{\left(}{\mathbb{R}}^n{\right)}}\ge\kappa_0$ for some constant $\kappa_0=\kappa_0{\left(}n,p,\Lambda{\right)}>0$.

By testing the inequality $-\Delta_{\operatorname{\overrightarrow{p}}}v\le\Lambda v^{p^*-1}$ with the 