Moreover, $\mathcal{I}_0\ne\emptyset$ due to $q_0<p_+$.

2.  For any $q>q_0$, there exists a constant $C_q=C{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u,q{\right)}$ such that $$\label{Th3Eq2}
    \left|u{\left(}x{\right)}\right|^{q}+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_q\bigg(1+\sum_{i\in\mathcal{I}_0^c}\left|x_i\right|^{\frac{qp_i}{q-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n,$$ where $\mathcal{I}_0^c:=\left\{1,\dotsc,n\right\}\backslash\mathcal{I}_0$.

We are able, moreover, to give an explicit definition in terms of $n$ and $\operatorname{\overrightarrow{p}}$ of a real number $q_0$ satisfying the above result (see Section \[Sec7\]).

The dependence on $u$ of the constants $C_0$, $C_q$, and $R_0$ in the above results will be made more precise in Remarks \[Rem2\] and \[Rem3\].

As a remark about the support of solutions, by a result in Vétois [@Vet2], we have that for any nonnegative solution $u$ of with $f{\left(}x,u{\right)}$ as in (see [@Vet2] for the general assumptions), if $u{\left(}x{\right)}=0$ for some $x\in{\mathbb{R}}^n$, then we have $u\equiv0$ on the affine subspace $\left\{y\in{\mathbb{R}}^n:\,y_i=x_i\quad\forall i=\left\{1,\dotsc,n\right\}\backslash\mathcal{I}_-\right\}$, where $\mathcal{I}_-$ is the set of all indices $i$ such that $p_i=\min{\left(}\left\{p_j\in\operatorname{\overrightarrow{p}}\right\}{\right)}$. In case all $p_i$ are equal to $p$, we obtain that either $u>0$ or $u\equiv0$, thus recovering the same result as Vazquez [@Vaz] found for the $p$–Laplace operator. In the presence of anisotropy, as shows for instance Theorem \[Th3\], this result does not hold true in general on the whole ${\mathbb{R}}^n$.

We also point out that in the limit case $p_+=p^*$, we are able to 