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 − Δ m u = u q | ∇ u | p in R N , where N ⩾ 1 , m > 1 and p , q ⩾ 0. The technique of Bernstein gradient estimates is utilized to study the case p < m. Moreover, a Liouville-type theorem for supersolutions under subcritical range of exponents q (N − m) + p (N − 1) < N (m − 1) 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 − Δ m u = f (x , u , ∇ 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. [ABSTRACT FROM AUTHOR]