  We prove a Liouville-type theorem that is one-dimensional symmetry and
classification results for non-negative $L^q$-viscosity solutions of the
equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)\pm
|Du|^p=0, x\in \mathbb{R}_+^n, \end{equation*} with boundary condition
$u(\tilde{x},0)=M\geq 0, \tilde{x}\in \mathbb{R}^{n-1}$, where
$\mathcal{M}_{\lambda, \Lambda}^{\pm}$ are the Pucci's operators with
parameters $\lambda, \Lambda \in \mathbb{R}_+$ $0<\lambda\leq \Lambda$ and
$p>1$. The results are an extension of the results by Porreta and Ver\'on in
arXiv:0805.2533 for the case $p\in (1,2]$ and by o Filippucci, Pucci and
Souplet in arXiv:1906.05161 for the case $p>2$, both for the Laplacian case
(i.e. $\lambda=\Lambda=1$). As an application in the case $p>2$, we prove a
sharp Bernstein estimation for $L^q$-viscosity solutions of the fully nonlinear
equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)=
|Du|^p+f(x), \quad x\in \Omega, \label{ecuacion1} \end{equation*} with boundary
condition $u=0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$.
