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This paper is concerned with two properties of positive weak solutions ofquasilinear elliptic equations with nonlinear gradient terms. First, we show aLiouville-type theorem for positive weak solutions of the equation involvingthe $m$-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 studythe case $p<m$. Moreover, a Liouville-type theorem for supersolutions undersubcritial 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 obtainthe existence of positive weak solutions for a nonlinear Dirichlet problem ofthe type $-\Delta_m u = f(x,u,\nabla u)$, with $f$ satisfying certain structureconditions. Our proof is based on a priori estimates, which will beaccomplished by using a blow-up argument together with the Liouville-typetheorem in the half-space. As another application, some new Harnackinequalities are proved.

Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms