  This paper is concerned with two properties of positive weak solutions of
quasilinear elliptic equations with nonlinear gradient terms. First, we show a
Liouville-type theorem for positive weak solutions of the equation involving
the $m$-Laplacian operator \begin{equation*} -\Delta_{m}u=u^q|\nabla u|^p\ \ \
\mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq1$, $m>1$ and
$p,q\geq0$. The technique of Bernstein gradient estimates is ultilized to study
the case $p<m$. Moreover, a Liouville-type theorem for supersolutions under
subcritial range of exponents \begin{equation*} q(N-m)+p(N-1)<N(m-1)
\end{equation*} is also established. Then, we use a degree argument to obtain
the existence of positive weak solutions for a nonlinear Dirichlet problem of
the type $-\Delta_m u = f(x,u,\nabla u)$, with $f$ satisfying certain structure
conditions. Our proof is based on a priori estimates, which will be
accomplished by using a blow-up argument together with the Liouville-type
theorem in the half-space. As another application, some new Harnack
inequalities are proved.
