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

Caihong Chang, Zhengce Zhang

Published 2020-08-17Version 1

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show the Liouville-type theorems 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 $p,q\geq0$. This paper mainly adopts the technique of Bernstein gradient estimates to study from three cases: $p>m$, $p=m$ and $p<m$, respectively. Meanwhile, this paper also obtains a Liouville-type theorem for supersolutions under certain assumptions on $m, p, q$ and $N$. Then, as its applications, we apply a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem. Meanwhile, some new Harnack inequalities are proved.