学术报告（张正策 12.28）

In this talk, we consider two properties of positive weak solutions of quasilinear elliptic equations, $-\Delta_{m}u=u^q|\nabla u|^p\ \mathrm{in}\ \mathbb{R}^N$, with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator. The technique of Bernstein gradient estimates is ultilized 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 $-\Delta_m u=f(x,u,\nabla u)$, 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. This is a joint work with Caihong Chang and Bei Hu.