gradient estimates, Namlyeyeva–Shishkov–Skrypnik [@NamShiSkr] on singular solutions, and Vétois [@Vet2] on vanishing properties of solutions. More references can be found for instance in [@Vet2].

Throughout this paper, we denote $$\label{Eq4}
p_+:=\max{\left(}\left\{p_i\in\operatorname{\overrightarrow{p}}\right\}{\right)}\quad\text{and}\quad p_*:=\frac{n-1}{\sum_{i=1}^n\frac{1}{p_i}-1}=\frac{p{\left(}n-1{\right)}}{n-p}\,.$$ The exponent $p_*$ is known to play a critical role in several results on the asymptotic behavior of solutions of second order elliptic equations (see the historic paper of Serrin [@Ser], see also for instance the more recent paper of Serrin–Zou [@SerZou] and the references therein).

Our first result is as follows.

\[Th1\] Assume that $p_+<p_*$. Let $f:{\mathbb{R}}^n\times{\mathbb{R}}\to{\mathbb{R}}$ be a Caratheodory function such that holds true and $u$ be a solution of . Then there exists a constant $C_0=C_0{\left(}n,\operatorname{\overrightarrow{p}},\Lambda,u{\right)}$ such that $$\label{Th1Eq1}
\left|u{\left(}x{\right)}\right|^{p_*}+\sum_{i=1}^n\left|\partial_{x_i}u{\left(}x{\right)}\right|^{p_i}\le C_0\bigg(1+\sum_{i=1}^n\left|x_i\right|^{\frac{p_*p_i}{p_*-p_i}}\bigg)^{-1}\quad\text{for a.e. }x\in{\mathbb{R}}^n,$$ where $p_*$ is as in .

We point out that the decay rate in is the same as the one obtained in Cîrstea–Vétois [@CirVet] for the fundamental solutions in ${\mathbb{R}}^n$, namely the solutions of the equation $-\Delta_{\operatorname{\overrightarrow{p}}}u=\delta_0$ in ${\mathbb{R}}^n$, where $\delta_0$ is the Dirac mass at the point 0.

In case all $p_i$ are equal to $p$, as part of a more general result, Alvino–Ferone–Trombetti–Lions [@AlvFerTroLio] proved that the best constant in the inequality is attained by the functions $$\label{Eq5}
u_{a,b}{\left(}x{\right)}:=\bigg(a+b\sum_{i=1}^n\left|x_i\right|^{\frac{p}{p-1}}\bigg)^{\frac{p-n}{p}}$$ for 